KNOW-HOW

imp and port connectors are sufficient for modeling most connections, but there is third connector, export, which is used exclusively in hierarchical connections.

Normally hierarchical routing is not required. A port on an UVC monitor can be connected directly to an imp in a scoreboard by specifying full hierarchical pathname (e.g., env.agent.monitor.analysis_port). The structure of an UVC is fixed and so the user knows to look in the monitor component for the analysis ports.

However the internal hierarchy of a module UVC is more arbitrary, and it may be convenient to route all the module UVC connectors to the top level to allow use without knowledge of the internal structure.

On the initiator side

• ports are routed up the hierarchy via other port instances

On the target side

• only the component which defines the communication method is allowed to have an imp instance. So we need a third object to route connections up the target side hierarchy - export

The hierarchical route must be connected at each level in the direction of control flow:

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initiator.connect(target)

Connection rules are as follows:

• port initiators can be connected to port, export or imp targets
• export initiators can be connected to export or imp targets
• imp cannot be connection initiator. imp is target only and is always the last connection object on a route

uvm_analysis_port

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class uvm_analysis_port # (
   	type 	T	 = 	int
) extends uvm_port_base # (uvm_tlm_if_base #(T,T))

uvm_analysis_port

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class uvm_analysis_port # (
   	type 	T	 = 	int
) extends uvm_port_base # (uvm_tlm_if_base #(T,T))

uvm_analysis_imp

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class uvm_analysis_imp #(
   	type 	T	 = 	int,
   	type 	IMP	 = 	int
) extends uvm_port_base #(uvm_tlm_if_base #(T,T))

QA

1. What are the three distinct functions of a scoreboard?

   Reference model, expected data storage and comparison

2. To how many consumer components can an analysis port be connected?

   Any number, including zero

3. What are the names of the two declarations made available by the following macro:

   1	`uvm_analysis_imp_decl(_possible)

   Subclass uvm_analysis_imp_possible

   Method write_possible

4. Why should a scoreboard clone the data received by a write method?

   The write method only sends a reference, therefore if the sending component uses the same reference for every data item, overwriting of data in the storage function of the scoreboard is possible without cloning.

compile

• -incdir for all UVC's `include
• all pkg.sv
• all if.sv
• all RTL DUT
• uvm top module

uvm top.sv

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module tb_top;
  
  // import the UVM library
  import uvm_pkg::*;

  // include the UVM macros
  `include "uvm_macros.svh"

  // import the yapp UVC
  import yapp_pkg::*;
  import hbus_pkg::*;
  import channel_pkg::*;
  import clock_and_reset_pkg::*;

  // include the test library file
  `include "router_tb.sv"
  `include "router_test_lib.sv"
    
  initial begin 
    yapp_vif_config::set(null, "uvm_test_top.tb.yapp.tx_agent.*", "vif", hw_top.in0);
    hbus_vif_config::set(null, "uvm_test_top.tb.hbus.*", "vif", hw_top.hif ); 
    channel_vif_config::set(null, "uvm_test_top.tb.chan0.*", "vif", hw_top.ch0);
    channel_vif_config::set(null, "uvm_test_top.tb.chan1.*", "vif", hw_top.ch1);
    channel_vif_config::set(null, "uvm_test_top.tb.chan2.*", "vif", hw_top.ch2);
    clock_and_reset_vif_config::set(null, "uvm_test_top.tb.clock_and_reset.*", "vif", hw_top.clk_rst_if);
    run_test();
  end


endmodule : tb_top

RTL top

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module hw_top;

  // Clock and reset signals
  logic [31:0]  clock_period;
  logic         run_clock;
  logic         clock;
  logic         reset;

  // YAPP Interface to the DUT
  yapp_if in0(clock, reset);
  // reset will now be generated by the Clock and Reset UVC
  // input: clock; output: reset, run_clock, clock_period
  clock_and_reset_if clk_rst_if(.clock(clock), .reset(reset), .run_clock(run_clock), .clock_period(clock_period));
  hbus_if hif(.clock(clock), .reset(reset));

  channel_if ch0(.clock(clock), .reset(reset));
  channel_if ch1(.clock(clock), .reset(reset));
  channel_if ch2(.clock(clock), .reset(reset));

  // CLKGEN module generates clock
  clkgen clkgen (
    .clock(clock),
    .run_clock(run_clock),
    .clock_period(clock_period)
  );

  yapp_router dut(
      ...


endmodule

pkg.sv

`include all related uvm class in pgk.sv

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package yapp_pkg;
import uvm_pkg::*;
`include "uvm_macros.svh"

typedef uvm_config_db#(virtual yapp_if) yapp_vif_config;

`include "yapp_packet.sv"
`include "yapp_tx_monitor.sv"
`include "yapp_tx_sequencer.sv"
`include "yapp_tx_seqs.sv"
`include "yapp_tx_driver.sv"
`include "yapp_tx_agent.sv"
`include "yapp_env.sv"

endpackage

if.sv

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interface yapp_if (input clock, input reset );
timeunit 1ns;
timeprecision 100ps;

import uvm_pkg::*;
`include "uvm_macros.svh"

import yapp_pkg::*;
    ...
    
endinterface : yapp_if

import and compile order

project

yapp_pkg.sv

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package yapp_pkg;
import uvm_pkg::*;
`include "uvm_macros.svh"

 typedef uvm_config_db#(virtual yapp_if) yapp_vif_config;

`include "yapp_packet.sv"
`include "yapp_tx_monitor.sv"
`include "yapp_tx_sequencer.sv"
`include "yapp_tx_seqs.sv"
`include "yapp_tx_driver.sv"
`include "yapp_tx_agent.sv"
`include "yapp_env.sv"

endpackage

yapp_tx_monitor.sv

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class yapp_tx_monitor extends uvm_monitor;
  
  // Collected Data handle
  yapp_packet pkt;

  // Count packets collected
  int num_pkt_col;

  // component macro
  `uvm_component_utils_begin(yapp_tx_monitor)
    `uvm_field_int(num_pkt_col, UVM_ALL_ON + UVM_NOCOMPARE)
  `uvm_component_utils_end
/////////////////// virtual interface ////////////////////
  virtual interface yapp_if vif;
/////////////////////////////////////////////////////////
      
  function new (string name, uvm_component parent);
    super.new(name, parent);
  endfunction : new
      ...
     
endclass : yapp_tx_monitor

yapp_tx_driver.sv

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class yapp_tx_driver extends uvm_driver #(yapp_packet);

 ////////////////// virtual interface ////////////////////
  virtual interface yapp_if vif;
 /////////////////////////////////////////////////////
      
  int num_sent;

  // component macro
  `uvm_component_utils_begin(yapp_tx_driver)
    `uvm_field_int(num_sent, UVM_ALL_ON + UVM_NOCOMPARE)
  `uvm_component_utils_end

  // Constructor - required syntax for UVM automation and utilities
  function new (string name, uvm_component parent);
    super.new(name, parent);
  endfunction : new

  function void start_of_simulation_phase(uvm_phase phase);
    `uvm_info(get_type_name(), {"start of simulation for ", get_full_name()}, UVM_HIGH)
  endfunction : start_of_simulation_phase
      ...
     
endclass : yapp_tx_driver 

yapp_if.sv

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interface yapp_if (input clock, input reset );
timeunit 1ns;
timeprecision 100ps;

import uvm_pkg::*;
`include "uvm_macros.svh"

////////////////// import package //////////////////////
import yapp_pkg::*;
//////////////////////////////////////////////////////    

  // Actual Signals
  logic              in_data_vld;
  logic              in_suspend;
  logic       [7:0]  in_data;

  // signal for transaction recording
  bit monstart, drvstart;

  // local storage for payload
  logic [7:0] payload_mem [0:63];
    ...
    
endinterface : yapp_if

virtual interface is in package;

yapp_pkg is imported into interface

!!! typedef uvm_config_db#(virtual yapp_if) yapp_vif_config; is forward declaration

A forward typedef declares an identifier as a type in advance of the full definition of that type

VCS compile

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vcs -full64 -R -sverilog  -ntb_opts uvm-1.2 +UVM_TESTNAME=short_yapp_012_test  -f vcs.f  

vcs.f

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-timescale=1ns/1ns

+incdir+../sv

../sv/yapp_if.sv
../sv/yapp_pkg.sv
./clkgen.sv
./hw_top.sv
./tb_top.sv

output ERROR

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Error-[SV-LCM-PND] Package not defined
../sv/yapp_if.sv, 18
yapp_if, "yapp_pkg::"
  Package scope resolution failed. Token 'yapp_pkg' is not a package. 
  Originating module 'yapp_if'.
  Move package definition before the use of the package.

xrun compile

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xrun -f xrun.f

xrun.f

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-64

-uvmhome  CDNS-1.2

// incdir for include files
-incdir ../sv

+UVM_TESTNAME=short_yapp_012_test

-timescale 1ns/1ns

// compile files
../sv/yapp_if.sv
../sv/yapp_pkg.sv
./clkgen.sv
./hw_top.sv
./tb_top.sv

output Error

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file: ../sv/yapp_if.sv
import yapp_pkg::*;
              |
xmvlog: *E,NOPBIND (../sv/yapp_if.sv,18|14): Package yapp_pkg could not be bound.
        interface worklib.yapp_if:sv
                errors: 1, warnings: 0

solution

place ../sv/yapp_pkg.sv before ../sv/yapp_if.sv.

In this particular example, yapp_pkg is NOT needed in interface. just delete import yapp_pkg::* is enough in ../sv/yapp_if.sv

plain example

The order of compilation unit DON'T matter

project

hw_top.sv

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module hw_top;

  // Clock and reset signals
  logic [31:0]  clock_period;
  logic         run_clock;
  logic         clock;
  logic         reset;

  // YAPP Interface to the DUT
  yapp_if in0(clock, reset);

  // CLKGEN module generates clock
  clkgen clkgen (
    .clock(clock),
    .run_clock(1'b1),
    .clock_period(32'd10)
  );

  yapp_router dut(
    .reset(reset),
    .clock(clock),
    .error(),
    // YAPP interface signals connection
    .in_data(in0.in_data),
    .in_data_vld(in0.in_data_vld),
    .in_suspend(in0.in_suspend),
    // Output Channels
    //Channel 0   
    .data_0(),
    .data_vld_0(),
    .suspend_0(1'b0),
    //Channel 1   
    .data_1(),
    .data_vld_1(),
    .suspend_1(1'b0),
    //Channel 2   
    .data_2(),  
    .data_vld_2(),
    .suspend_2(1'b0),
    // Host Interface Signals
    .haddr(),
    .hdata(),
    .hen(),
    .hwr_rd());
    
    ...
        
endmodule

yapp_router.sv

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module yapp_router (input clock,                              
                    input reset,                            
                    output error,

                    // Input channel
                    input [7:0] in_data,                           
                    input in_data_vld,                     
                    output in_suspend,
                    // Output Channels
                    output [7:0] data_0,  //Channel 0
                    output reg data_vld_0, 
                    input suspend_0, 
                    output [7:0] data_1,  //Channel 1
                    output reg data_vld_1, 
                    input suspend_1, 
                    output [7:0] data_2,  //Channel 2
                    output reg data_vld_2,
                    input suspend_2,
     
                    // Host Interface Signals
                    input [15:0] haddr,
                    inout [7:0] hdata,
                    input hen,
                    input hwr_rd);                            
...
endmodule    

compile

place hw_top.sv before or after yapp_router.sv doesn't matter, the compiler (xrun, vcs) can compile them successfully.

Conclusion

Always place package before DUT is preferred choice during compiling.

refernce

class forward declaration

https://vlsiverify.com/uvm/uvm_config_db-in-uvm#Precedence_Rule

Two Precedence rules

There are two precedence rules applicable to uvm_config_db. In the build_phase,

1. A set() call in a context higher up the component hierarchy takes precedence over a set() call that occurs lower in the hierarchical path.
2. On having same context field, the last set() call takes precedence over the earlier set() call.

Example for rule - 1

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`include "uvm_macros.svh"
import uvm_pkg::*;

class component_A extends uvm_component;
  int id;
  `uvm_component_utils(component_A)
  
  function new(string name = "component_A", uvm_component parent = null);
    super.new(name, parent);
    id = 1;
  endfunction
  
  function display();
    `uvm_info(get_type_name(), $sformatf("inside component_A: id = %0d", id), UVM_LOW);
  endfunction
endclass

class component_B extends component_A;
  int receive_value;
  int id;
  
  `uvm_component_utils(component_B)
  
  function new(string name = "component_B", uvm_component parent = null);
    super.new(name, parent);
    id = 2;
  endfunction
  
  function void build_phase(uvm_phase phase);
    super.build_phase(phase);
    if(!uvm_config_db #(int)::get(this, "*", "value", receive_value))
      `uvm_fatal(get_type_name(), "get failed for resource in this scope");    
  endfunction
  
  function display();
    `uvm_info(get_type_name(), $sformatf("inside component_B: id = %0d, receive_value = %0d", id, receive_value), UVM_LOW);
  endfunction
endclass

class env extends uvm_env;
  `uvm_component_utils(env)
  component_A comp_A;
  component_B comp_B;
  
  function new(string name = "env", uvm_component parent = null);
    super.new(name, parent);
  endfunction
  
  function void build_phase(uvm_phase phase);
    super.build_phase(phase);
    comp_A = component_A ::type_id::create("comp_A", this);
    comp_B = component_B ::type_id::create("comp_B", this);
    
    uvm_config_db #(int)::set(this, "*", "value", 200);
  endfunction
  
  task run_phase(uvm_phase phase);
    super.run_phase(phase);
    void'(comp_A.display());
    void'(comp_B.display());
  endtask
endclass

class my_test extends uvm_test;
  bit control;
  `uvm_component_utils(my_test)
  env env_o;
  
  function new(string name = "my_test", uvm_component parent = null);
    super.new(name, parent);
  endfunction
  
  function void build_phase(uvm_phase phase);
    super.build_phase(phase);
    ////////////////////////////////////////////////////
      env_o = env::type_id::create("env_o", this);				// line 99
    
      uvm_config_db #(int)::set(null, "*", "value", 100);		// line 101
	////////////////////////////////////////////////////
  endfunction
   
  function void end_of_elaboration_phase(uvm_phase phase);
    super.end_of_elaboration_phase(phase);
    uvm_top.print_topology();
  endfunction
endclass

module tb_top;
  initial begin
    run_test("my_test");
  end
endmodule

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UVM_INFO /xcelium20.09/tools//methodology/UVM/CDNS-1.2/sv/src/base/uvm_root.svh(605) @ 0: reporter [UVMTOP] UVM testbench topology:
--------------------------------------
Name          Type         Size  Value
--------------------------------------
uvm_test_top  my_test      -     @1810
  env_o       env          -     @1877
    comp_A    component_A  -     @1922
    comp_B    component_B  -     @1953
--------------------------------------

UVM_INFO testbench.sv(14) @ 0: uvm_test_top.env_o.comp_A [component_A] inside component_A: id = 1
UVM_INFO testbench.sv(36) @ 0: uvm_test_top.env_o.comp_B [component_B] inside component_B: id = 2, receive_value = 100

As you can see , the uvm_config_db::set is after the factory type_id::create, but receive_value is still 100, I believe that build_phase of upper hierarchy finish first before create lower components.

But the recommendation in literature and many web is place uvm_config_db::set before create

Simple Example for rule - 2

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uvm_config_db #(int)::set(null, "*", "value", 100);
uvm_config_db #(int)::set(null, "*", "value", 200);

[complete code]

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UVM_INFO /xcelium20.09/tools//methodology/UVM/CDNS-1.2/sv/src/base/uvm_root.svh(605) @ 0: reporter [UVMTOP] UVM testbench topology:
--------------------------------------
Name          Type         Size  Value
--------------------------------------
uvm_test_top  my_test      -     @1810
  env_o       env          -     @1877
    comp_A    component_A  -     @1924
    comp_B    component_B  -     @1955
--------------------------------------

UVM_INFO testbench.sv(14) @ 0: uvm_test_top.env_o.comp_A [component_A] inside component_A: id = 1
UVM_INFO testbench.sv(36) @ 0: uvm_test_top.env_o.comp_B [component_B] inside component_B: id = 2, receive_value = 200

Another Example for rule - 2

The set in super.build_phase and set in current build_phase are at the same hierarchy, so rule-2 apply

base_test

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class base_test extends uvm_test;

  // component macro
  `uvm_component_utils(base_test)

  router_tb tb;

  // component constructor
  function new(string name, uvm_component parent);
    super.new(name, parent);
  endfunction : new

  // UVM build_phase()
  function void build_phase(uvm_phase phase);
    super.build_phase(phase);
    uvm_config_int::set(this, "*", "recording_detail", 1);
    uvm_config_wrapper::set(this, "tb.yapp.tx_agent.sequencer.run_phase", 
                                                      "default_sequence",
                                                      yapp_5_packets::get_type());
    tb = router_tb::type_id::create("tb", this);
  endfunction : build_phase

endclass : base_test

test2 - One

yapp_incr_payload_seq override the yapp_5_packets in super.build_phase(phase), yapp_incr_payload_seq is used.

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class test2 extends base_test;

  // component macro
  `uvm_component_utils(test2)

  // component constructor
  function new(string name, uvm_component parent);
    super.new(name, parent);
  endfunction : new

  function void build_phase(uvm_phase phase);
    yapp_packet::type_id::set_type_override(short_yapp_packet::get_type());
    super.build_phase(phase);
    uvm_config_wrapper::set(this, "tb.yapp.tx_agent.sequencer.run_phase", 
                                                      "default_sequence",
                                                      yapp_incr_payload_seq::get_type());
  endfunction : build_phase

endclass : test2

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---------------------------------------------------------------------------------------------------------------------                   
Name                           Type               Size  Value
---------------------------------------------------------------------------------------------------------------------                   
req                            short_yapp_packet  -     @3045
  length                       integral           6     'h5
  addr                         integral           2     'h1
  payload                      da(integral)       5     -
    [0]                        integral           8     'h0
    [1]                        integral           8     'h1
    [2]                        integral           8     'h2
    [3]                        integral           8     'h3
    [4]                        integral           8     'h4
  parity                       integral           8     'h11
  parity_type                  parity_t           1     GOOD_PARITY
  packet_delay                 integral           32    'd19
  begin_time                   time               64    0
  depth                        int                32    'd2
  parent sequence (name)       string             21    yapp_incr_payload_seq                                                           
  parent sequence (full name)  string             61    uvm_test_top.tb.yapp.tx_agent.sequencer.yapp_incr_payload_seq                   
  sequencer                    string             39    uvm_test_top.tb.yapp.tx_agent.sequencer                                         
---------------------------------------------------------------------------------------------------------------------

test2 - Two

the yapp_5_packets in super.build_phase(phase) overide yapp_incr_payload_seq in test2's build_phase, yapp_5_packets is used

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class test2 extends base_test;

  // component macro
  `uvm_component_utils(test2)

  // component constructor
  function new(string name, uvm_component parent);
    super.new(name, parent);
  endfunction : new

  function void build_phase(uvm_phase phase);
    yapp_packet::type_id::set_type_override(short_yapp_packet::get_type());
    uvm_config_wrapper::set(this, "tb.yapp.tx_agent.sequencer.run_phase", 
                                                      "default_sequence",
                                                      yapp_incr_payload_seq::get_type());
    super.build_phase(phase);
  endfunction : build_phase

endclass : test2

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--------------------------------------------------------------------------------------------------------------                          
Name                           Type               Size  Value
--------------------------------------------------------------------------------------------------------------                          
req                            short_yapp_packet  -     @3176
  length                       integral           6     'h6
  addr                         integral           2     'h1
  payload                      da(integral)       6     -
    [0]                        integral           8     'h4f
    [1]                        integral           8     'hbe
    [2]                        integral           8     'hb
    [3]                        integral           8     'h1d
    [4]                        integral           8     'h72
    [5]                        integral           8     'hc5
  parity                       integral           8     'h49
  parity_type                  parity_t           1     GOOD_PARITY
  packet_delay                 integral           32    'd0
  begin_time                   time               64    40
  depth                        int                32    'd2
  parent sequence (name)       string             14    yapp_5_packets                                                                  
  parent sequence (full name)  string             54    uvm_test_top.tb.yapp.tx_agent.sequencer.yapp_5_packets                          
  sequencer                    string             39    uvm_test_top.tb.yapp.tx_agent.sequencer                                         
-------------------------------------------------------------------------------------------------------------- 

• Laplace transform
  • a generalization of the continuous-time Fourier transform
  • converts integro-differential equations into algebraic equations
• z-transforms
  • a generalization of the discrete-time Fourier transform
  • converts difference equations into algebraic equations

system function, transfer function: \(H(s)\)

frequency response: \(H(j\omega)\), if the ROC of \(H(s)\) includes the imaginary axis, i.e.\(s=j\omega \in \text{ROC}\)

Laplace Transform

To specify the Laplace transform of a signal, both the algebraic expression and the ROC are required. The ROC is the range of values of \(s\) for the integral of \(t\) converges

bilateral Laplace transform

where \(s\) in the ROC and \(\mathfrak{Re}\{s\}=\sigma\)

The formal evaluation of the integral for a general \(X(s)\) requires the use of contour integration in the complex plane. However, for the class of rational transforms, the inverse Laplace transform can be determined without directly evaluating eq. (9.56) by using the technique of partial fraction expansion

ROC Property

The range of values of s for which the integral in converges is referred to as the region of convergence (which we abbreviate as ROC) of the Laplace transform

i.e. no pole in RHP for stable LTI sytem

System Causality

For a causal LTI system, the impulse response is zero for \(t \lt 0\) and thus is right sided

causality implies that the ROC is to the right of the rightmost pole, but the converse is not in general true, unless the system function is rational

System Stability

The system is stable, or equivalently, that \(h(t)\) is absolutely integrable and therefore has a Fourier transform, then the ROC must include the entire \(j\omega\)-axis

all of the poles have negative real parts

Unilateral Laplace transform

analyzing causal systems and, particularly, systems specified by linear constant-coefficient differential equations with nonzero initial conditions (i.e., systems that are not initially at rest)

A particularly important difference between the properties of the unilateral and bilateral transforms is the differentiation property \(\frac{d}{dt}x(t)\)

Integration by parts for unilateral Laplace transform

in Bilateral Laplace Transform

In fact, the initial- and final-value theorems are basically unilateral transform properties, as they apply only to signals \(x(t)\) that are identically \(0\) for \(t \lt 0\).

\(z\)-Transform

The \(z\)-transform for discrete-time signals is the counterpart of the Laplace transform for continuous-time signals

where \(z=re^{j\omega}\)

The \(z\)-transform evaluated on the unit circle corresponds to the Fourier transform

ROC Property

system stability

The system is stable, or equivalently, that \(h[n]\) is absolutely summable and therefore has a Fourier transform, then the ROC must include the unit circle

\(z\)-Transform Table

[http://www.ws.binghamton.edu/Fowler/fowler%20personal%20page/EE301_files/ZT%20Tables_rev2.pdf]

Unilateral \(z\)-Transform

The time shifting property is different in the unilateral case because the lower limit in the unilateral transform definition is fixed at zero, \(x[n-n_0]\)

bilateral \(z\)-transform

unilateral \(z\)-transform

Initial rest condition

Initial Value Theorem & Final Value Theorem

Laplace Transform

Two valuable Laplace transform theorem

• Initial Value Theorem, which states that it is always possible to determine the initial value of the time function \(f(t)\) from its Laplace transform \[ \lim _{s\to \infty}sF(s) = f(0^+) \]

• Final Value Theorem allows us to compute the constant steady-state value of a time function given its Laplace transform \[ \lim _{s\to 0}sF(s) = f(\infty) \]

  If \(f(t)\) is step response, then \(f(0^+) = H(\infty)\) and \(f(\infty) = H(0)\), where \(H(s)\) is transfer function

\(z\)-transform

Initial Value Theorem \[ f[0]=\lim_{z\to\infty}F(z) \] final value theorem \[ \lim_{n\to\infty}f[n]=\lim_{z\to1}(z-1)F(z) \]

Coert Vonk. Initial/final value proofs [https://coertvonk.com/physics/lfz-transforms/z/initial-final-value-proofs-31543]

\(s\)- and \(z\)-Domains Connection

discrete-time systems also can be analyzed by means of the Laplace transform — the \(z\)-transform is the Laplace transform in disguise and that discrete-time systems can be analyzed as if they were continuous-time systems.

A continuous-time system with transfer function \(H(s)\) that is identical in structure to the discrete-time system \(H[z]\) except that the delays in \(H[z]\) are replaced by elements that delay continuous-time signals.

delay element	Time domain	Transform
\(z\)-transform	\(\delta[n-1]\)	\(z^{-1}\)
Laplace transform	\(\delta(t-T)\)	\(e^{-sT}\)

Therefore, we obtain

\[\begin{align} \sum_{n=0}^{\infty}y[n]z^{-n} &=\sum_{n=0}^{\infty}x[n]z^{-n}\cdot\sum_{n=0}^{\infty}h[n]z^{-n} \\ \sum_{n=0}^{\infty}y[n]e^{-snT} &=\sum_{n=0}^{\infty}x[n]e^{-snT}\cdot \sum_{n=0}^{\infty}h[n]e^{-snT} \end{align}\]

The \(z\)-transform of a sequence can be considered to be the Laplace transform of impulses sampled train with a change of variable \(z = e^{sT}\) or \(s = \frac{1}{T}\ln z\)

Note that the transformation \(z = e^{sT}\) transforms the imaginary axis in the \(s\) plane (\(s = j\omega\)) into a unit circle in the \(z\) plane (\(z = e^{sT} = e^{j\omega T}\), or \(|z| = 1\))

Note: \(\bar{h}(t)\) is the impulse sampled version of \(h(t)\)

Note \(\bar{h}(t)\) is impulse sampled signal, whose CTFT is scaled by \(\frac{1}{T}\) of continuous signal \(h(t)\), \(\overline{H}[e^{sT}]=\overline{H}(e^{sT})\) is the approximation of continuous time system response, for example summation \(\frac{1}{1-z^{-1}}\) \[ \frac{1}{1-z^{-1}} = \frac{1}{1-e^{-sT}} \approx \frac{1}{j\omega \cdot T} \] And we know transform of integral \(u(t)\) is \(\frac{1}{s}\), as expected there is ratio \(T\)

Staszewski, Robert Bogdan, and Poras T. Balsara. All-digital frequency synthesizer in deep-submicron CMOS. John Wiley & Sons, 2006

D. Sundararajan. Signals and Systems A Practical Approach Second Edition, 2023

impulse invariance

A straightforward and useful relationship between the continuous-time impulse response \(h_c(t)\) and the discrete-time impulse response \(h[n]\)

\[ h[n] = Th_c(nT) \]

and \(T\) is chosen such that

\[ H_c(j\omega)=0, \space\space |\hat{\omega}| \ge \frac{\pi}{T} \]

When \(h[n]\) and \(h_c(t)\) are related through the above equation, i.e., the impulse response of the discrete-time system is a scaled, sampled version of \(h_c(t)\), the discrete-time system is said to be an impulse-invariant version of the continuous-time system

we have \[ H(e^{j\hat{\omega}}) = H_c\left(j\frac{\hat{\omega}}{T}\right),\space\space |\hat{\omega}| \lt \pi \]

• \(h[n] = Th_c(nT)\) and \(T\to 0\)

  guarantees \(y_c(nT) = y_r(nT)\), i.e. output equivalence only at the sampling instants

• \(H_c(j\Omega)\) is bandlimited and \(T \lt 1/2f_h\)

  guarantees \(y_c(t) = y_r(t)\)

The scaling of \(T\) can alternatively be thought of as a normalization of the time domain, that is average impulse response shall be same i.e., \(h[n]\times 1 = h(nT)\times T\)

note that the impulse-invariant transform is not appropriate for high-pass responses, because of aliasing errors

⭐ bandlimited

⭐ bandlimited

Transfer function & sampled impulse response

continuous-time filter designs to discrete-time designs through techniques such as impulse invariance

useful functions

• using fft

  The outputs of the DFT are samples of the DTFT

• using freqz

  modeling as FIR filter, and the impulse response sequence of an FIR filter is the same as the sequence of filter coefficients, we can express the frequency response in terms of either the filter coefficients or the impulse response

  fft is used in freqz internally

freqz method is straightforward, which apply impulse invariance criteria. Though fft is used for signal processing mostly,

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clear all;
clc;

%% continuous system
s = tf('s');
h = 2*pi/(2*pi+s); 	% First order lowpass filter with 3-dB frequency 1Hz
[mag, phs, wout] = bode(h);
fct = wout(:)/2/pi;
Hct_dB = 20*log10(mag(:));


fstep = 0.01;           % freq resolution
fnyqst = 1000;
Ts = 1/(2*fnyqst);
Fs = 1/Ts;              % sampling freq
Ns = ceil(Fs/fstep);    % samping points
fstep = Fs/Ns;          % update fstep
t = (0:Ns-1)*Ts;        % sampling time points

y = impulse(h, t);      % impulse resp

%% modelling as discrete system
Y = fft(y);                 % dft
Hfft = Y * Ts;              % !!! multiply Ts
Hfft_dB = 20*log10(abs(Hfft(1:Ns/2+1)));
ffft = (1:Ns/2+1)*fstep - fstep;


[Hfir, ffir] = freqz(y, 1, [], 1/Ts);   % modelling as FIR
Hfir = Hfir * Ts;           % !!! multiply Ts
Hfir_dB = 20*log10(abs(Hfir));

%% plot
semilogx(fct, Hct_dB, '-ok', ffft, Hfft_dB, 'r.-', ffir, Hfir_dB, 'b--');
legend('bode(s)', 'fft', 'FIR model')
xlabel('Freq(Hz)');
ylabel('dB');
xlim([1e-2 1e2]);
grid on;
title('frequency response of different methods');

!!! only multiply Ts

FIR Equalization

Frequency Response

\[ z = e^{j\omega T_s} \]

Unit impulse

filter coefficients are [-0.131, 0.595, -0.274] and sampling period is 100ps

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%% Frequency response
w = [-0.131, 0.595, -0.274];
Ts = 100e-12;
[mag, w] = freqz(w, 1, [], 1/Ts);
subplot(2, 1, 1)
plot(w/1e9, 20*log10(abs(mag)));
xlabel('Freq(GHz)');
ylabel('dB');
grid on;
title('frequency response');

%% unit impulse response from transfer function
subplot(2, 1, 2)
z = tf('z', Ts);
h = -0.131 + 0.595*z^(-1) -0.274*z^(-2);
[y, t] = impulse(h);
stem(t*1e10, y*Ts);  % !!! y*Ts is essential
grid on;
title("unit impulse response");
xlabel('Time(\times 100ps)');
ylabel('mag');

impulse:

For discrete-time systems, the impulse response is the response to a unit area pulse of length Ts and height 1/Ts, where Ts is the sample time of the system. (This pulse approaches \(\delta(t)\) as Ts approaches zero.)

Scale output:

Multiply impulse output with sample period Ts in order to correct 1/Ts height of impulse function.

PSD transformation

If we have power spectrum or power spectrum density of both edge's absolute jitter (\(x(n)\)) , \(P_{\text{xx}}\)

Then 1UI jitter is \(x_{\text{1UI}}(n)=x(n)-x(n-1)\), and Period jitter is \(x_{\text{Period}}(n)=x(n)-x(n-2)\), which can be modeled as FIR filter, \(H(\omega) = 1-z^{-k}\), i.e. \(k=1\) for 1UI jitter and \(k=2\) Period jitter \[\begin{align} P_{\text{xx}}'(\omega) &= P_{\text{xx}}(\omega) \cdot \left| 1-z^{-k} \right|^2 \\ &= P_{\text{xx}}(\omega) \cdot \left| 1-(e^{j\omega T_s})^{-k} \right|^2 \\ &= P_{\text{xx}}(\omega) \cdot \left| 1-e^{-j\omega T_s k} \right|^2 \end{align}\]

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clear all
close all
clc

xf_fs = 0:0.01:0.5;
k = 1;
H_1UI = 1 - exp(-1i*2*pi*xf_fs);
HH_1UI = abs(H_1UI).^2;
subplot(2, 1, 1);
plot(xf_fs, HH_1UI);
grid on;
xlabel('Freq');
ylabel('|H|^2')
title('Weight for 1UI jitter');

k = 2;
H_period = 1 - exp(-1i*2*pi*xf_fs);
HH_period = abs(H_period).^2;
subplot(2, 1, 2)
plot(xf_fs, HH_period);
grid on;
xlabel('Freq');
ylabel('|H|^2')
title('Weight for Period jitter');

\[ x(t-\Delta T)\overset{FT}{\longrightarrow} X(s)e^{-\Delta T \cdot s} \]

Bilinear Transform

[https://tttapa.github.io/Pages/Mathematics/Systems-and-Control-Theory/Digital-filters/Discretization/Bilinear-transform.html]

an algebraic transformation between the variables \(s\) and \(z\) that maps the entire imaginary \(j\Omega\)-axis in the \(s\)-plane to one revolution of the unit circle in the \(z\)-plane

\[\begin{align} z &= \frac{1+s\frac{T_s}{2}}{1-s\frac{T_s}{2}} \\ s &= \frac{2}{T_s}\cdot \frac{1-z^{-1}}{1+z^{-1}} \end{align}\]

where \(T_s\) is the sampling period

frequency warping:

• The bilinear transformation avoids the problem of aliasing encountered with the use of impulse invariance, because it maps the entire imaginary axis of the \(s\)-plane onto the unit circle in the \(z\)-plane

• impulse invariance cannot be used to map highpass continuous-time designs to high pass discrete-time designs, since highpass continuous-time filters are not bandlimited

• Due to nonlinear warping of the frequency axis introduced by the bilinear transformation, bilinear transformation applied to a continuous-time differentiator will not result in a discrete-time differentiator. However, impulse invariance can be applied to bandlimited continuous-time differentiator

  The feature of the frequency response of discrete-time differentiators is that it is linear with frequency

The simple approximation \(z=e^{sT}\approx1+sT\), the first equal come from impulse invariance essentially

Stephen Roberts. Lecture 3 - Design of Digital Filters [https://www.robots.ox.ac.uk/~sjrob/Teaching/B14_SP/b14_sp_lect3.pdf]

Design of IIR Filters From Analog Filters

Modeling a Continuous-Time System with Matlab

Because the mapping between the continuous (\(s\)-plane) and discrete domains (\(z\)-plane) cannot be done exactly, the various design methods are at best approximations

Neil Robertson. Modeling a Continuous-Time System with Matlab [https://www.dsprelated.com/showarticle/1055.php]

—. Modeling Anti-Alias Filters [https://www.dsprelated.com/showarticle/1418.php]

impinvar [https://www.mathworks.com/help/signal/ref/impinvar.html]

bilinear [https://www.mathworks.com/help/signal/ref/bilinear.html]

filter [https://www.mathworks.com/help/matlab/ref/filter.html]

---

The design of IIR filters [https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/14f29be83ac38f11eb1a1bd5fdb6f5ea_lecture_18.pdf]

The design of IIR filters (cont.) [https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/cc00ac6d468dc9dcf2238fc1d1a194d4_lecture_19.pdf]

Lecture 19: Design of IIR Filters [http://smartdata.ece.ufl.edu/eee5502/2020_fall/media/2020_eee5502_slides28.pdf]

Approximation of Derivatives

Perhaps the simplest method for low-order systems is to use backward-difference approximation to continuous domain derivatives

Note this approximation is not same with impulse invariance, e.g. \(\frac{1}{s^3} \to \frac{T^3z(z+1)}{2(-1)^3}\) employing impulse invariance

\[ 1- z^{-1} = 1-e^{j\Omega T} = 1-\cos(\omega T) + j\sin(\Omega T) \approx 1-1+j\Omega T = s T \]

That is

\[ s \approx \frac{1-z^{-1}}{T} \]

Matched z-Transform (Root Matching)

The matched Z-transform method, also called the pole-zero mapping or pole-zero matching method, and abbreviated MPZ or MZT

\[ z = e^{sT} \]

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% https://www.dsprelated.com/showarticle/1642.php

%II One-pole RC filter model
Ts= 1/fs;
Wc= 1/(R*C);               % rad -3 dB frequency
fc= Wc/(2*pi);             % Hz -3 dB frequency
a1= -exp(-Wc*Ts);
b0= 1 + a1;                % numerator coefficient
a= [1 a1];                 % denominator coeffs
y_filt= filter(b0,a,y);    % filter the DAC's output signal y

Impulse Invariance (impinvar)

• To extend the accurate frequency range, we would need to increase the sample rate
• impulse-invariant transform is not appropriate for high-pass responses, because of aliasing errors

Note \(h[n] = Th_c(nT)\), Multiply the analog impulse response by this gain to enable meaningful comparison (other response, like step, the amplitude correction is not needed)

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% https://www.dsprelated.com/showarticle/1055.php
% modified by hguo, Sun Jun 22 09:21:48 AM CST 2025

% butter_3rd_order.m      6/4/17 nr
% Starting with the butterworth transfer function in s,
% Create discrete-time filter using the impulse invariance xform and compare
% its time and frequency responses to those of the continuous time filter.
% Filter fc = 1 rad/s = 0.159 Hz


% I.  Given H(s), find H(z) using the impulse-invariant transform
fs= 4;            % Hz  sample frequency
% 3rd order butterworth polynomial
num= 1;
den= [1 2 2 1];
[b,a]= impinvar(num,den,fs);         % coeffs of H(z)
%[b,a]= bilinear(num,den,fs)

% II.  Impulse Response and Step Response
% find discrete-time impulse response
Ts= 1/fs;
N= 16*fs;
n= 0:N-1;
t= n*Ts;
x= [1, zeros(1,N-1)];   % impulse
x= fs*x;              % make impulse response amplitude independent of fs
y= filter(b,a,x);    % filter the impulse  
subplot(2,2,1),plot(t,y,'.'),grid,
xlabel('seconds')

% Continuous-time Impulse response from inverse Laplace transform
% Blinchikoff and Zverev, p116
h= exp(-t) - 2/sqrt(3)*exp(-t/2).*cos(sqrt(3)/2*t + pi/6); 
subplot(2,2,2),plot(t,y,'.',t,h),grid
title('Impulse Response'),legend('discrete-time filter response', 'continuous-time filter response'),xlabel('seconds')

% find discrete-time step response
x= ones(1,N);         % step
y= filter(b,a,x);    % filter the step         
% Continuous-time step response. Blinchikoff and Zverev, p116
t= t+Ts/2;                 % offset t to to align step responses
h= 1 - exp(-t) - 2/sqrt(3)*exp(-t/2).*sin(sqrt(3)/2*t); 
e= h-y;                    % error of discrete-time response
subplot(2,2,3),plot(t,y,'.',t,h),grid
title('Step Response'),legend('discrete-time filter response', 'continuous-time filter response'),xlabel('seconds')
% discrete-time response error
subplot(2,2,4),plot(t,e),grid
title('Error of discrete-time step response'),xlabel('seconds')

• To use the filter function with the b coefficients from an FIR filter, use y = filter(b,1,x).

Bilinear Transformation (bilinear)

TODO 📅

bandwidth from pulse response

L.W. Couch, Digital and Analog Communication Systems, 8th Edition, Pearson, 2013.

Generating Basic signals – Rectangular Pulse and Power Spectral Density using FFT

Convolution Property of the Fourier Transform \[ x(t)*h(t)\longleftrightarrow X(\omega)H(\omega) \] pulse response can be obtained by convolve impulse response with UI length rectangular \[ H(\omega) = \frac{Y_{\text{pulse}}(\omega)}{X_{\text{rect}}(\omega)} = \frac{Y_{\text{pulse}}(\omega)}{\text{sinc}(\omega)} \]

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% Convolution Property of the Fourier Transform
% pulse(t) = h(t) * rect(t)
% -> fourier transform
% PULSE = H * RECT
% FT(RECT) = sinc
% H = PULSE/RECT = PULSE/sinc
xx = pi*ui.*w(1:plt_num);
y_sinc = ui.*sin(xx)./xx;
y_sinc(1) = y_sinc(2);
y_sinc = y_sinc/y_sinc(1);  % we dont care the absoulte gain
h_ban1 = abs(h(1:plt_num))./abs(y_sinc);

h_dB = 20*log10(abs(h_ban1));
[hmin, Index] = min(abs(h_dB +3 ));
f_3dB = w(Index);
f_3dB = f_3dB/1e9;

sinc function

Notice that the complete definition of \(\operatorname{sinc}\) on \(\mathbb R\) is \[ \operatorname{Sa}(x)=\operatorname{sinc}(x) = \begin{cases} \frac{\sin x}{x} & x\ne 0, \\ 1, & x = 0, \end{cases} \] which is continuous.

To approach to real spectrum of continuous rectangular waveform, \(\text{NFFT}\) has to be big enough.

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clear;
close all;
clc;

fs=500; %sampling frequency
Ts = 1/fs;
T=0.2; %width of the rectangule pulse in seconds

figure(1)
hold on;
t=-0.5:1/fs:0.5; %time base
x=rectpuls(t,T); %generating the square wave
sum(x>0.5)
stem(t,x,'--k');
plot(t, x, 'b.-')
xstart = T/2-Ts/2;
xend = -T/2-Ts/2;
plot([xstart, xstart], [-1, 1.02], 'r--');
plot([xend, xend], [-1, 1.02], 'r--');
plot([-0.11, 0.11], [1, 1], 'r--')
hold off;
grid on;
title(['Rectangular Pulse width=', num2str(T),'s']);
xlabel('Time(s)');
ylabel('Amplitude');
xlim([-0.12, 0.12]);
ylim([-0.05, 1.05]);

figure(2)
hold on
Titer = [1, 4, 16, 32, 64, 128];
color = {'k', 'g', 'r', 'm', 'c', 'b'};
formatSpec = 'NFFT=%d';
for k=1:length(Titer)
    t=-Titer(k):Ts:Titer(k); %time base
    x=rectpuls(t,T); %generating the square wave

    L=length(x);
    Np = nextpow2(L)-1;
    NFFT = 2^Np;
    X = fftshift(fft(x,NFFT)); %FFT with FFTshift for both negative & positive frequencies
    f = fs*(-NFFT/2:NFFT/2-1)/NFFT; %Frequency Vector
    Xc = abs(X)*Ts; % continuous signal spectrum
    plot(f, Xc, "Color",color{k}, 'LineWidth',2);
    legend_str{k} = [num2str(NFFT, formatSpec)];
end
hold off
legend(legend_str);
title('Magnitude of X_c (original continuous signal)');
xlabel('Frequency (Hz)')
ylabel('Magnitude |X(f)|');
grid on;

reference

Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab. 1996. Signals & systems (2nd ed.)

Alan V Oppenheim, Ronald W. Schafer. 2010. Discrete-Time Signal Processing, 3rd edition

B.P. Lathi, Roger Green. Linear Systems and Signals (The Oxford Series in Electrical and Computer Engineering) 3rd Edition

Sam Palermo, ECEN720, Lecture 7: Equalization Introduction & TX FIR Eq

Sam Palermo, ECEN720, Lab5 –Equalization Circuits

B. Razavi, "The z-Transform for Analog Designers [The Analog Mind]," IEEE Solid-State Circuits Magazine, Volume. 12, Issue. 3, pp. 8-14, Summer 2020. [https://www.seas.ucla.edu/brweb/papers/Journals/BR_SSCM_3_2020.pdf]

Jhwan Kim, CICC 2022, ES4-4: Transmitter Design for High-speed Serial Data Communications

Mathuranathan. Digital filter design – Introduction [https://www.gaussianwaves.com/2020/02/introduction-to-digital-filter-design/]

Daniel Boschen, "Fast Track to Designing FIR Filters with Python" [https://events.gnuradio.org/event/24/contributions/598/attachments/186/485/Boschen%20FIR%20Filter%20Presentation.pdf]

Daniel Boschen, "Quick Start on Control Loops with Python" [https://events.gnuradio.org/event/24/contributions/599/attachments/187/480/Boschen%20Control%20Presentation.pdf]

Most user-defined UVM classes are placed in separate files

Classes can only be compiled from a module or package scope

Recommendation:

• Include related class files into a package
• Import the package into modules where required

The UVM library is supplied in the package uvm_pkg

• Import package to access library
• The macro file uvm_macros.svh must be included separately

yapp_packet.sv

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class yapp_packet extends uvm_sequence_item;
    ...
endclass : yapp_packet   

yapp_pkg.sv

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package  yapp_pkg;
	import uvm_pkg::*;
	`include "uvm_macros.svh"

	`include "yapp_packet.sv"
endpackage : yapp_pkg

run.f for xrun

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// 64 bit option for AWS labs
-64

 -uvmhome CDNS-1.2

// include directories
-incdir ../sv

// compile files
../sv/yapp_pkg.sv
./top.sv

Matching network

TODO 📅

Q factor

TODO 📅

Self-Resonant Frequency

\[ f_\text{SRF} = \frac{1}{2\pi \sqrt{LC}} \] The SRF of an inductor is the frequency at which the parasitic capacitance of the inductor resonates with the ideal inductance of the inductor, resulting in an extremely high impedance. The inductance only acts like an inductor below its SRF

• For choking applications, chose an inductor whose SRF is at or near the frequency to be attenuated

• For other applications, the SRF should be at least 10 times higher than the operating frequency

  it is more important to have a relatively flat inductance curve (constant inductance vs. frequency) near the required frequency

[Understanding RF Inductor Specifications, https://www.ece.uprm.edu/~rafaelr/inel5325/SupportDocuments/doc671_Selecting_RF_Inductors.pdf]

[RFIC-GPT Wiki, https://wiki.icprophet.net/]

T-coil

• Broaden the bandwidth
• Create a constant, resistive input impedance in the presence of a heavy load capacitance (ESD protection circuit)

L₁ = L₂ for impedance matching

The use of T-coils can dramatically increase the bandwidth and improve the return loss in both TXs and RXs.

equivalent circuit

Note the negative sign of \(L_M\), which is a consequence of magnetic coupling; owing to this, the driving impedance at the center tap as seen by \(C\) is lower than without the coupling.

\[\begin{align} Z_{c} &= sL_a/2 -sL_M = sL/2 - sL_M/2 \\ Z_{noc} &= sL/2 \end{align}\]

Bridged T-Coil

The vertical stacking of the inductor halves causes a significant bridging capacitance \(C_B\)​ between them.

The most important aspect is that \(C_B\) does not prohibit perfect impedance matching but can be used to tune certain transfer functions.

Tcoil in RX

ppwl: Independent Piece-Wise Linear Resistive Source

simple model

L1, L2, Km, Cb

lumped model

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simulator lang=spectre
include "/path/to/INDL0.scs"
subckt for_import(p0 p1 p2 gnd)
	xmod (p0 p1 p2 gnd) INDL0
ends for_import

x_1 (p_1_1 p_1_2 p_1_3 0) for_import
v_1_1 (p_1_1 0) vsource mag=-1
v_1_2 (p_1_2 0) vsource mag=0
v_1_3 (p_1_3 0) vsource mag=0
save v_1_1:p v_1_2:p v_1_3:p

x_2 (p_2_1 p_2_2 p_2_3 0) for_import
v_2_1 (p_2_1 0) vsource mag=0
v_2_2 (p_2_2 0) vsource mag=-1
v_2_3 (p_2_3 0) vsource mag=0
save v_2_1:p v_2_2:p v_2_3:p

x_3 (p_3_1 p_3_2 p_3_3 0) for_import
v_3_1 (p_3_1 0) vsource mag=0
v_3_2 (p_3_2 0) vsource mag=0
v_3_3 (p_3_3 0) vsource mag=-1
save v_3_1:p v_3_2:p v_3_3:p
Y ac start=1.000000e+08 stop=2.000000e+10 step=1.000000e+08

xsp (p_1 p_2 p_3 0) for_import
port1 (p_1 0) port
port2 (p_2 0) port
port3 (p_3 0) port
S sp start=1.000000e+08 stop=2.000000e+10 step=1.000000e+08 ports=[ port1 port2 port3]

EMX_plot_tcoil in emxform.ils

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(define (EMX_plot_tcoil bgui wid what)
  (needNCells 'adtComplex 100000)
  (letseq ((to_z (lambda (ys)
		   (letseq ((y11 (EMX_matrix_ref ys 0 0))
			    (y12 (EMX_matrix_ref ys 0 1))
			    (y21 (EMX_matrix_ref ys 1 0))
			    (y22 (EMX_matrix_ref ys 1 1))
			    (det y11*y22-y12*y21)
			    (z11 y22/det)
			    (z12 -y12/det)
			    (z22 y11/det))
		     (list z11 z12 z22))))
	   (ground
	    (lambda (y p)
	      (letseq ((n (EMX_matrix_rows y))
		       (yg (make_EMX_matrix n-1)))
		(do ((i 0 i+1))
		    ((i >= n-1))
		  (do ((j 0 j+1))
		      ((j >= n-1))
		    (EMX_matrix_set yg i j (EMX_matrix_ref y i+(if (i >= p) 1 0) j+(if (j >= p) 1 0)))))
		yg)))
	   (reduce
	    (lambda (yys)
	      (letseq ((xvec (drGetWaveformXVec (car yys)))
		       (n (drVectorLength xvec))
		       (yyvecs (mapcar drGetWaveformYVec yys))
		       (zz1 (drCreateVec 'doublecomplex n))
		       (zz2 (drCreateVec 'doublecomplex n))
		       (zz3 (drCreateVec 'doublecomplex n)))
		(do ((i 0 i+1))
		    ((i >= n))
		  (let ((ys (mapcar (lambda (w) (drGetElem w i)) yyvecs)))
		    (setq ys (ground (as_EMX_matrix 3 3 ys) 2))
		    (let ((zz (to_z ys)))
		      (drSetElem zz1 i (nth 0 zz))
		      (drSetElem zz2 i (nth 1 zz))
		      (drSetElem zz3 i (nth 2 zz)))))
		(list (drCreateWaveform xvec zz1)
		      (drCreateWaveform xvec zz2)
		      (drCreateWaveform xvec zz3)))))
	   (get_k
	    (lambda (l12 l1122)
	      (letseq ((xvec (drGetWaveformXVec l12))
		       (n (drVectorLength xvec))
		       (l12v (drGetWaveformYVec l12))
		       (l1122v (drGetWaveformYVec l1122))
		       (resultv (drCreateVec 'double n)))
		(do ((i 0 i+1))
		    ((i >= n))
		  (letseq ((l12i (drGetElem l12v i))
			   (l1122i (drGetElem l1122v i))
			   (kk (if (l1122i > 0.0)
				   l12i/(sqrt l1122i)
				   0.0))
			   (k (if ((abs kk) < 2.0) kk 0.0)))
		    (drSetElem resultv i k)))
		(drCreateWaveform xvec resultv)))))
  (EMX_plot_aux bgui wid what 3
		'("Inductance" "Q" "k")
		'("Henry" "" "")
		(lambda (ys)
		  (letseq ((zs (reduce ys))
			   (z11 (nth 0 zs))
			   (z12 (nth 1 zs))
			   (z22 (nth 2 zs))
			   (pi 3.14159265358979)
			   (f (xval z11))
			   (l11 (imag z11)/(2*pi*f))
			   (q11 (imag z11)/(real z11))
			   (l12 (imag z12)/(2*pi*f))
			   (l22 (imag z22)/(2*pi*f))
			   (q22 (imag z22)/(real z22))
			   (k (get_k l12 l11*l22)))
		    `((,l11 ,l22) (,q11 ,q22) (,k))))
		'(("L1" "L2") ("Q1" "Q2") ("k")))))

Tcoil vs tapped inductor

tcoil and tapped inductor share same EM simulation result, and use modelgen with different model formula.

The relationship is \[ L_{\text{sim}} = L1_{\text{sim}}+L2_{\text{sim}}+2\times k_{\text{sim}} \times \sqrt{L1_{\text{sim}}\cdot L2_{\text{sim}}} \] where \(L1_{\text{sim}}\), \(L2_{\text{sim}}\) and \(k_{\text{sim}}\) come from tcoil model result, \(L_{\text{sim}}\) comes from tapped inductor model result

\(k_{\text{sim}}\) in EMX have assumption, induce current from P1 and P2 Given Dot Convention:

Same direction : k > 0

Opposite direction : k < 0

So, the \(k_{\text{sim}}\) is negative if routing coil in same direction

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% EMX - shield tcoil model
L1 = csvread('./L1sim.csv', 1, 0);
L2 = csvread('./L2sim.csv', 1, 0);
k = csvread('./ksim.csv', 1, 0);

% EMX - Tapped shield inductor
L = csvread('./Lsim.csv', 1, 0);

freq = L1(:, 1)/1e9;    % GHz
L1 = L1(:, 2);
L2 = L2(:, 2);
k = -k(:, 2);   % Caution: minus of EMX ksim due to same current direction

L = L(:, 2);

Lcalc = L1 + L2 + 2*k.*(L1.*L2).^0.5;

plot(freq, L*1e9, 'r', 'LineWidth', 3);
hold on;
plot(freq, Lcalc*1e9, '--b', 'LineWidth', 3);
grid on;
xlabel('Freq (GHz)');
ylabel('Inductance (nH)');
legend('Tapped inductor model', 'tcoil model calc');

电磁感应定律 (electromagnetic induction)

任何封闭电路中感应电动势大小，等于穿过这一电路磁通量的变化率。 \[ \epsilon = -\frac{d\Phi_B}{dt} \] 其中 \(\epsilon\)是电动势，单位为伏特

\(\Phi_B\)是通过电路的磁通量，单位为韦伯

电动势的方向（公式中的负号）由楞次定律决定

楞次定律: 由于磁通量的改变而产生的感应电流，其方向为抗拒磁通量改变的方向。

在回路中产生感应电动势的原因是由于通过回路平面的磁通量的变化，而不是磁通量本身，即使通过回路的磁通量很大，但只要它不随时间变化，回路中依然不会产生感应电动势。

自感电动势

当电流\(I\)随时间变化时，在线圈中产生的自感电动势为 \[ \epsilon = -L\frac{dI}{dt} \]

同名端：当两个电流分别从两个线圈的对应端子流入 ，其所 产生的磁场相互加强时，则这两个对应端子称为同名端。

reference

S. Shekhar, J. S. Walling and D. J. Allstot, "Bandwidth Extension Techniques for CMOS Amplifiers," in IEEE Journal of Solid-State Circuits, vol. 41, no. 11, pp. 2424-2439, Nov. 2006

David J. Allstot Bandwidth Extension Techniques for CMOS Amplifiers

B. Razavi, "The Bridged T-Coil [A Circuit for All Seasons]," IEEE Solid-State Circuits Magazine, Volume. 7, Issue. 40, pp. 10-13, Fall 2015.

B. Razavi, "The Design of Broadband I/O Circuits [The Analog Mind]," IEEE Solid-State Circuits Magazine, Volume. 13, Issue. 2, pp. 6-15, Spring 2021.

S. Galal and B. Razavi, "Broadband ESD protection circuits in CMOS technology," in IEEE Journal of Solid-State Circuits, vol. 38, no. 12, pp. 2334-2340, Dec. 2003, doi: 10.1109/JSSC.2003.818568.

M. Ker and Y. Hsiao, "On-Chip ESD Protection Strategies for RF Circuits in CMOS Technology," 2006 8th International Conference on Solid-State and Integrated Circuit Technology Proceedings, 2006, pp. 1680-1683, doi: 10.1109/ICSICT.2006.306371.

M. Ker, C. Lin and Y. Hsiao, "Overview on ESD Protection Designs of Low-Parasitic Capacitance for RF ICs in CMOS Technologies," in IEEE Transactions on Device and Materials Reliability, vol. 11, no. 2, pp. 207-218, June 2011, doi: 10.1109/TDMR.2011.2106129.

Bob Ross, "T-Coil Topics" DesignCon IBIS Summit 2011

Ross, Bob and Cong Ling. “Wang Algebra: From Theory to Practice.” IEEE Open Journal of Circuits and Systems 3 (2022): 274-285.

S. Lin, D. Huang and S. Wong, "Pi Coil: A New Element for Bandwidth Extension," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 56, no. 6, pp. 454-458, June 2009

Starič, Peter and Erik Margan. “Wideband amplifiers.” (2006).

Kosnac, Stefan (2021) Analysis of On-Chip Inductors and Arithmetic Circuits in the Context of High Performance Computing [https://archiv.ub.uni-heidelberg.de/volltextserver/30559/1/Dissertation_Stefan_Kosnac.pdf]

Chapter 4.5. High Frequency Passive Devices [https://www.cambridge.org/il/files/7713/6698/2369/HFIC_chapter_4_passives.pdf]

Inertial delay

Inertial delay models are simulation delay models that filter pulses that are shorted than the propagation delay of Verilog gate primitives or continuous assignments (assign #5 y = ~a;)

​ COMBINATIONAL LOGIC ONLY !!!

• Inertial delays swallow glitches
• sequential logic implemented with procedure assignments DON'T follow the rule

continuous assignments

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`timescale 1ns/100ps
module tb;

	reg in;
/////////////////////////////////////////////////////
	wire out;
	assign #2.5 out = in;
/////////////////////////////////////////////////////
	initial begin
		in = 0;
		#16;
		in = 1;
		#2;
		in = 0;
		#10;
		in = 1;
		#4;
		in = 0;
	end

	initial begin
		#50;
		$finish();
	end

endmodule

procedure assignment - combinational logic

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`timescale 1ns/100ps
module tb;

	reg in;
	reg out;

//////////// combination logic ////////////////////////
	always @(*)
		#2.5 out = in;
///////////////////////////////////////////////////////
/* the above code is same with following code
    	always @(*) begin
			#2.5;
			out = in;
		end
*/
	initial begin
		in = 0;
		#16;
		in = 1;
		#2;
		in = 0;
		#10;
		in = 1;
		#4;
		in = 0;
	end

	initial begin
		#50;
		$finish();
	end

endmodule

procedure assignment - sequential logic

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`timescale 1ns/100ps
module tb;
	reg clk;

	reg in;
	reg out;

	always begin
		clk = 0;
		#5;
		forever begin
			clk = ~clk;
			#5;
		end
	end
//////////// sequential logic //////////////////
	always @(posedge clk)
		#2.5 out <= in;
///////////////////////////////////////////////
	initial begin
		in = 0;
		#16;
		in = 1;
		#2;
		in = 0;
		#10;
		in = 1;
	end

	initial begin
		#50;
		$finish();
	end

endmodule

As shown above, sequential logic DON'T follow inertial delay

Transport delay

Transport delay models are simulation delay models that pass all pulses, including pulses that are shorter than the propagation delay of corresponding Verilog procedural assignments

• Transport delays pass glitches, delayed in time
• Verilog can model RTL transport delays by adding explicit delays to the right-hand-side (RHS) of a nonblocking assignment

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always @(*)
    y <= #5 ~a;

nonblocking assignment

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`timescale 1ns/100ps
module tb;

	reg in;
	reg out;
/////////////// nonblocking assignment ///
	always @(*) begin
		out <= #2.5 in;
	end
/////////////////////////////////////////
	initial begin
		in = 0;
		#16;
		in = 1;
		#2;
		in = 0;
		#10;
		in = 1;
		#4;
		in = 0;
	end

	initial begin
		#50;
		$finish();
	end

endmodule

blocking assignment

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`timescale 1ns/100ps
module tb;

	reg in;
	reg out;
/////////////// blocking assignment ///
	always @(*) begin
		out = #2.5 in;
	end
/////////////////////////////////////////
	initial begin
		in = 0;
		#16;
		in = 1;
		#2;
		in = 0;
		#10;
		in = 1;
		#4;
		in = 0;
	end

	initial begin
		#50;
		$finish();
	end

endmodule

It seems that new event is discarded before previous event is realized.

reference

Verilog Nonblocking Assignments With Delays, Myths & Mysteries

Correct Methods For Adding Delays To Verilog Behavioral Models

Article (20488135) Title: Selecting Different Delay Modes in GLS (RAK) URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O3w000009bdLyEAI

Article (20447759) Title: Gate Level Simulation (GLS): A Quick Guide for Beginners URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1Od0000005xEorEAE

What Is Real Number Modeling?

• model analog blocks operation as signal flow model
• only the digital solver is used for high-speed simulation
  • Event-driven
  • No convergence issues, because no analog solver is used
• Five different language standards support real number modeling:
  • wreal (wired-real) ports in Verilog-AMS
  • real data type in VHDL
  • real data type in Verilog
  • real variables and nettypes in SystemVerilog (SV)
  • real types in e

Benefits of RNM

• Most analog circuits that need to be modeled for MS verification at the SoC level can be described in terms of real-valued voltages or currents
• RNM is a mixed approach, borrowing concepts from both continuous and discrete domains
  • The values are floating-point (real) number.
  • Time is discrete; the real signals change values based on discrete events
• Applicability of RNM is bounded primarily by signal-flow model style
• Migrating analog behavior from the analog domain to the event or pseudo-analog domain can bring huge benefits without sacrificing too much accuracy
• Simulation is executed by a digital simulation engine without need for the analog solver
• Hence real-number modeling enables very high performance simulation of mixed-signal systems

Limitations of RNM

• connecting real or wreal signals to electrical signals requires careful consideration
  • Too conservative an approach can lead to large numbers of timepoints
  • Too liberal an approach can lead to losing signal accuracy
• Time accuracy limited by the discrete sampling approach and the `timescale setting - no continuous signals anymore
• Limited capability for combination of signals by wiring outputs together
  • Requires assumptions about impedances to do simple merging

LVS check

LVS issue for circuits with customized devices

• auCdl: Analog and Microwave CDL, is a netlister used for creating CDL netlist for analog circuits

• auLVS: Analog and Microwave LVS, is used for analog circuit LVS

EMX ports

plain labels

• pin layer
• uncheck Cadence pins in Advanced options

rectangle pins

• drawing layer rectangle pin and specify Access Direction as intended
• check Cadence pins in Advanced options

The rectangle pins are always selected as driven port while there are only rectangle pin whether Cadence pins checked or not.

check ports used for simulation

use GDS view - EMX

EMX Synthesis Kits

Synthesis is a capability of the EMX Pcell library and uses scalable model data pre-generated by Continuum for a specific process and metal scheme combination.

Synthesis is supported by the Pcells that are suffixed _scalable, and these Pcells have the additional fields and buttons needed for synthesis.

port order (signals)

emxform.ils

EMX device info

EMX plot function

EMX's formulation is defined in

1	/path/to/EMX/share/emx/virtuoso_ui/emxinterface/emxskill/emxform.ils

EMX import this file at Virtuoso startup, you have to relaunch Virtuoso if you change this file

Single-ended inductor

Both with and without shield apply

• port-1 impedance when port-2 short

\[ Z_1 = \frac{1}{Y_{11}} \]

• port-2 impedance when port-1 short

\[ Z_2 = \frac{1}{Y_{22}} \] Then \[\begin{align} L1 &= \frac{Im(Z_1)}{2\pi f} \\ Q1 &= \frac{Im(Z_1)}{Re(Z_1)} \\ L2 &= \frac{Im(Z_2)}{2\pi f} \\ Q2 &= \frac{Im(Z_2)}{Re(Z_2)} \end{align}\]

EMX only plot L1 and Q1

differential impedance

Y parameters to Z parameters

\[\begin{align} |Y| &= Y_{11}*Y_{22} - Y_{12}*Y_{22} \\ \begin{bmatrix} Z_{11} & Z_{12}\\ Z_{21} & Z_{22} \end{bmatrix} &= \begin{bmatrix} \frac{Y_{22}}{|Y|} & \frac{-Y_{12}}{|Y|}\\ \frac{-Y_{21}}{|Y|} & \frac{Y_{11}}{|Y|} \end{bmatrix} \end{align}\]

Then differential impedance is \[ Z_{diff} = Z_{11} - Z_{12} - Z_{21} + Z_{22} \]

similarly, Z parameters to Y parameters \[ \begin{bmatrix} Y_{11} & Y_{12}\\ Y_{21} & Y_{22} \end{bmatrix} = \begin{bmatrix} \frac{Z_{22}}{|Z|} & \frac{-Z_{12}}{|Z|}\\ \frac{-Z_{21}}{|Z|} & \frac{Z_{11}}{|Z|} \end{bmatrix} \] where \[ |Z| = Z_{11}Z_{22} - Z_{12}Z_{21} \]

Differential inductor

Both with and without shield apply

\[\begin{align} L_{diff} &= \frac{Im(Z_{diff})}{2\pi f} \\ Q_{diff} &= \frac{Im(Z_{diff})}{Re(Z_{diff})} \end{align}\]

Center-tapped inductor

\[ Y = \begin{bmatrix} Y_{11} & Y_{12} & Y_{13}\\ Y_{21} & Y_{22} & Y_{23}\\ Y_{31} & Y_{32} & Y_{33} \end{bmatrix} \]

where port order is P1 P2 CT.

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(define (EMX_plot_ct_ind bgui wid what)
  (EMX_plot_aux bgui wid what 3
		'("Differential inductance" "Differential Q")
		'("Henry" "")
		(lambda (ys)
		  (letseq ((z (EMX_differential (nth 0 ys) (nth 1 ys) (nth 3 ys) (nth 4 ys)))
			   (f (xval z))
			   (L (imag z)/(2*3.14159265358979*f))
			   (Q (imag z)/(real z)))
		    `((,L) (,Q))))
		'(("L") ("Q"))))

Assume CT i.e. port 3 in S-parameter is grounded, (z (EMX_differential (nth 0 ys) (nth 1 ys) (nth 3 ys) (nth 4 ys))) obtain differential impedance with \(Y_{11}\), \(Y_{12}\), \(Y_{21}\) and \(Y_{22}\). \[ Y = \begin{bmatrix} Y_{11} & Y_{12}\\ Y_{21} & Y_{22} \end{bmatrix} \] Finally, differential inductance and Q are obtained, shown as below

\[\begin{align} L_{diff} &= \frac{Im(Z_{diff})}{2\pi f} \\ Q_{diff} &= \frac{Im(Z_{diff})}{Re(Z_{diff})} \end{align}\]

Single-ended cap

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(define (EMX_plot_se_cap bgui wid what)
  (EMX_plot_aux bgui wid what 2
		'("Capacitance" "Q" "Capacitance" "Q")
		'("Farad" "" "Farad" "")
		(lambda (ys)
		  (letseq ((z1 1.0/(nth 0 ys))
			   (y12 (nth 1 ys))
			   (z2 1.0/(nth 3 ys))
			   (f (xval z1))
			   (C1 (-1.0/(imag z1))/(2*3.14159265358979*f))
			   (C12 -(imag y12)/(2*3.14159265358979*f))
			   (C2 (-1.0/(imag z2))/(2*3.14159265358979*f))
			   (Q1 -(imag z1)/(real z1))
			   (Q12 (imag y12)/(real y12))
			   (Q2 -(imag z2)/(real z2)))
		    `((,C1) (,Q1) (,C12))))
		'(("Cse") ("Qse") ("C12"))))

We define Port-1 impedance \(Z_1\), Port-2 impedance \(Z_2\)

\[\begin{align} Z_1 &= \frac {1}{Y_{11}}\\ Z_2 &= \frac {1}{Y_{22}} \end{align}\]

Then single-ended cap and Q \[\begin{align} C_1 &= -\frac{1/Im(Z_1)}{2\pi f} \\ Q_1 &= -\frac{Im(Z_1)}{Re(Z_1)} \\ C_2 &= -\frac{1/Im(Z_2)}{2\pi f} \\ Q_2 &= -\frac{Im(Z_2)}{Re(Z_2)} \\ C_{12} &= -\frac{Im(Y_{12})}{2\pi f}\\ Q_{12} &= \frac{Im(Y_{12})}{Re(Y_{12})} \end{align}\]

• Series equivalent model is used in \(C_1\), \(Q_1\), \(C_2\) and \(Q_2\)
  • \(Z_1 = R + \frac{1}{sC_1}\) and \(Z_2 = R + \frac{1}{sC_2}\)
• Parallel model is used in \(C_{12}\) and \(Q_{12}\)
  • \(Y_{12} = \frac{1}{R} + sC_{12}\)

EMX plot \(C_{se}\), \(Q_{se}\) and \(C_{12}\), i.e. \(C_1\), \(Q_1\) and \(C_{12}\)

Differential cap

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(define (EMX_plot_diff_cap bgui wid what)
  (EMX_plot_aux bgui wid what 2
		'("Differential capacitance" "Differential Q")
		'("Farad" "")
		(lambda (ys)
		  (letseq ((z (apply EMX_differential ys))
			   (f (xval z))
			   (C (-1.0/(imag z))/(2*3.14159265358979*f))
			   (Q -(imag z)/(real z)))
		    `((,C) (,Q))))
		'(("C") ("Q"))))

First obtain differential impedance, \(Z_{diff}\) then apply series equivalent model \[\begin{align} C_{diff} &= -\frac{1/Im(Z_{diff})}{2\pi f} \\ Q_{diff} &= -\frac{Im(Z_{diff})}{Re(Z_{diff})} \end{align}\]

Tline

Open circuit impedance \(Z_o\), short circuit impedance \(Z_s\) and characteristic impedance \(Z_0\)

\[\begin{align} Z_o &= Z_{11}\\ Z_s &= \frac{1}{Y_{11}}\\ Z_0 &= \sqrt{Z_o*Z_s} \end{align}\]

propagation constant is given as \[\begin{align} \gamma &= \frac{1}{2}\log\left( \frac{Z_0+Z_s}{Z_0-Z_s} \right) \\ &= \alpha + j\beta \end{align}\] where \(\alpha\) is attenuation constant and \(\beta\) is phase constant

The relationship between these parameter and geometry of the transmission line \[\begin{align} Z_0 &= \sqrt{\frac{R+j\omega L}{G+j\omega C}} \\ \gamma &= \sqrt{(G+j\omega C)(R+j\omega L)} \end{align}\] EMX plot the real and imaginary part of \(Z_0\), \(\alpha\) and \(\beta\) of \(\gamma\)

Note EMX plot the absolute value of \(\alpha\) and \(\beta\)

Transformer

[IC Prophet GDSII 文件使用和仿真测试说明]

EMX autoplot

using AC simulation, and inductor's parallel model or series model

That is to say: both sp (network parameter) and ac (impedance) can be used to plot inductance, Q value.

usually EMX choose ac method

left 2 figures are used for AC simulation, \(Y_{nn}\) can be obtained conveniently

Foundary model

for single-end capicator \[\begin{align} Q_1 &= -\frac{Im(Z_1)}{Re(Z_1)} \\ &= -\frac{Im(1/Y_{11})}{Re(1/Y_{11})} \\ &= -\frac{Im(Y_{11}^*)/|Y_{11}|^2}{Re(Y_{11}^*)/|Y_{11}|^2} \\ &= \frac{Im(Y_{11})}{Re(Y_{11})} \end{align}\]

So, the EMX model and foundary model is consistent.

Tips

Process file encryption mostly for advanced nodes, like TSMC 16nm Finfet, whose process file is encrypted.

• Use --key=EMXkey in the EMX Advanced options

GDSviewer has two options

• EMX: shows the final gds sent to EMX for simulation after it has been processed by EMX
• Raw: shows the raw gds

If there are port name with the # sign, it means EMX sees a port but it is not in the signal list.

EMX Accuracy

• Edge mesh: controls layout discretization in the X-Y plane

  • For MoM capacitors, use the edge mesh to be the same as the width of the finger (for example, 0.1um).

• Thickness: controls layout discretization in the Z dimension

• 3D metals: skips all 2D assumptions about conductors and their currents and charges

  • If you set 3D metals to * then all metals are treated as 3D
    • For Inductor type structures, only thick metal needs 3D.
    • For MoM, all layers are needed.

Ports entered in Grounds will cause these nets to be grounded; these ports will not show up in the S-parameter result.

Setup Temperature

• EMX: --temperature=100

ParaView

• If check ParaView related options when ParaView is not setup properly, EMX simulation stop at Creating mesh... without waring or errors (version 6.2).

pad & bump

EMX process file contain M0 up to RDL-AP

PM, CB2_FC, UBM is in the chip package

PEX extract up to RDL-AP as expected

DC model

quick find routing resistance

GPDK045 metal resistor model is not consistent with its process file on sheet resistance

reference

Tips on Specifying Ports in EMX [link]

Using 'Cadence pins' as ports with access direction in EMX simulations [[link](Article (20496398) Title: Using 'Cadence pins' as ports with access direction in EMX simulations URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O3w00000AH2OfEAL)]

EMX miscellaneous features [https://picture.iczhiku.com/resource/eetop/WyIFKleSLTRIuvCb.pdf]