Isolation cells are additional cells inserted by the synthesis tools
for isolating the buses/wires crossing from power-gated
domain of a circuit to its always-on domain
(AON).
To prevent corruption of always-on domain, we clamp the nets crossing
the power domains to a value depending upon the design.
A simple circuit having a switchable (or gated) power
domain
The circuit shown in Figure 1, after isolation cells are
inserted
Jitter separation lets you learn if the components of jitter are
random or deterministic. That is, if they are caused by crosstalk,
channel loss, or some other phenomenon. The identification of jitter and
noise sources is critical when debugging failure sources in the
transmission of high-speed serial signals
Tail Fit Method
Spectral method
RJ Extraction Methods
Rationale
Spectral
Speed/Consistency to Past Measurements; Accuracy in low
Crosstalk or Aperiodic Bounded Uncorrelated Jitter (ABUJ)
conditions
Tail Fit
General Purpose; Accuracy in high Crosstalk or ABUJ
conditions
power spectral density (PSD) represents jitter spectrum and peaks in
the spectrum can be interpreted as PJ or DDJ, while the average noise
floor is the power of RJ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
S1 = sum(win); S2 = sum(win.^2); N = length(win); spec_nospur2 = (spec_nospur*S1).^2/N/S2; % To obtain linear spectrum for rj rj_utj = sqrt(sum(spec_nospur2))*1e12;
spec = 1*ones(length(spec_nospur), 1)*1e-21; spec(index) = specx(index); % insert fft nyquist frequency component between positive frequency and % negative frequency component % DC;posFreq;nyqFreq;negFreq spec_ifft = [spec;specnyq;conj(spec(end:-1:2))]'; sfactor = sum(win)/sqrt(2); spec_ifft = spec_ifft*sfactor; sig_rec = real(ifft(spec_ifft)); sig_rec = sig_rec(:); sig_rec_utj = sig_rec./win(1:end);
Tail Fit Method
Tail fitting algorithm based on the Gaussian tail
model by using probability distribution of collected jitter
value
1 2 3 4 5 6 7 8 9 10 11 12 13
bin_sig = bin_sig*1e12;
x = qfuncinv(cdf_sig);
% coef(1)*bin_sig + coef(2) = x % which x is norm(0, 1) % bin_sig = (x - coef(2))/coef(1) % Then bin is norm(-coef(2)/coef(1), 1/coef(1)) coef = polyfit(bin_sig, x, 1); sigma = 1/coef(1); mu = -coef(2)*sigma;
fprintf('sigma=%.3fps, mu=%.3fps\n', sigma, mu);
Least Squares (LS) method
It is known that TIE jitter is a linear equation, shown in below
formula \[
x[n] = d_n \times \left[ \Delta t_{pj}[n]+\Delta t_{DCD}[n] +\Delta
t_{ISI}[n]+\Delta t_{RJ}[n]\right]
\] LS can be used to estimate the PJ, DCD, RJ , and ISI
parameters \([a,b,J_{DCD},J_0,
J_1...J_{(2^k-1)}]\)
Jitter modeling
Periodic Jitter (PJ)
PJ is a repeating jitter \[
\Delta t_{PJ}[n]=A\sin(2\pi f_0\cdot nT_s + \theta)=a \sin(2\pi f_0
\cdot nT_s)+b\cos(2\pi f_0 \cdot nT_s)
\] where \(f_0\) represents the
fundamental frequency of PJ; \(A\) is
the amplitude of PJ; \(T_s\) is the
data stream period, and \(\theta\) is
the initial phase of PJ
In the spectrum, the frequency of maximum amount of the jitter is PJ
frequency \(f_0\).
Duty Cycle Distortion (DCD)
DCD is viewed as a series of adjacent positive and negative
impulses\[
\Delta t_{DCD}[n] = J_{DCD}\times (-1)^n =
[-J_{DCD},J_{DCD},-J_{DCD},J_{DCD},...]
\] Where \(J_{DCD}\) is the DCD
amplitude.
Random Jitter (RJ)
RJ is created by unbounded jitter sources, such as Gaussian white
noise. The statistical PDF for RJ is enerally treated as a Gaussian
distribution \[
f_{RJ}(\Delta t) = \frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{(\Delta
t)^2}{2\sigma^2})
\]
Remarks
Periodic Jitter
Generator and Insertion
Analysis and Estimation of Jitter Sub-Components: Classification
and Segregation of Jitter Components
Reference
Mike Li. 2007. Jitter, noise, and signal integrity at high-speed
(First. ed.). Prentice Hall Press, USA.
Y. Duan and D. Chen, "Accurate jitter decomposition in high-speed
links," 2017 IEEE 35th VLSI Test Symposium (VTS), 2017, pp. 1-6, doi:
10.1109/VTS.2017.7928918.
Y. Duan and D. Chen, "Fast and Accurate Decomposition of
Deterministic Jitter Components in High-Speed Links," in IEEE
Transactions on Electromagnetic Compatibility, vol. 61, no. 1, pp.
217-225, Feb. 2019, doi: 10.1109/TEMC.2018.2797122.
"Jitter Analysis: The Dual-Dirac Model, RJ/DJ, and Q-Scale",
Whitepaper: Keysight Technologies, U.S.A., Dec. 2017
Sharma, Vijender Kumar and Sujay Deb. "Analysis and Estimation of
Jitter Sub-Components." (2014).
Qingqi Dou and J. A. Abraham, "Jitter decomposition in ring
oscillators," Asia and South Pacific Conference on Design Automation,
2006
E. Balestrieri, L. De Vito, F. Lamonaca, F. Picariello, S. Rapuano
and I. Tudosa, "The jitter measurement ways: The jitter decomposition,"
in IEEE Instrumentation & Measurement Magazine, vol. 23, no. 7, pp.
3-12, Oct. 2020, doi: 10.1109/MIM.2020.9234759.
McClure, Mark Scott. "Digital jitter measurement and separation." PhD
diss., 2005.
Ren, Nan, Zaiming Fu, Shengcu Lei, Hanglin Liu, and Shulin Tian.
"Jitter generation model based on timing modulation and cross point
calibration for jitter decomposition." Metrology and Measurement Systems
28, no. 1 (2021).
M. P. Li, J. Wilstrup, R. Jessen and D. Petrich, "A new method for
jitter decomposition through its distribution tail fitting,"
International Test Conference 1999. Proceedings (IEEE Cat.
No.99CH37034), 1999, pp. 788-794, doi: 10.1109/TEST.1999.805809.
K. Bidaj, J. -B. Begueret and J. Deroo, "RJ/DJ jitter decomposition
technique for high speed links," 2016 IEEE International Conference
on Electronics, Circuits and Systems (ICECS) [https://sci-hub.se/10.1109/ICECS.2016.7841269]
BTW, the psd value at half of fundamental frequency (\(f_s/2\)) is duty cycle distortion due
to the NMOS/PMOS imbalance, because of rising only
data
Random Jitter
RJ can be accurately and efficiently measured using
PSS/Pnoise or HB/HBnoise.
Note that the transient noise can also be used to
compute RJ;
However, the computation cost is typically very high, and the
accuracy is lesser as compared to PSS/Pnoise and HB/HBnoise.
Since RJ follows a Gaussian distribution, it can be fully
characterized using its Root-Mean-Squared value (RMS) or the standard
deviation value (\(\sigma\))
The Peak-to-Peak value of RJ (\(\text{RJ}_{\text{p-p}}\)) can be calculated
under certain observation conditions \[
\text{RJ}_{\text{p-p}}\equiv K \ast \text{RJ}_{\text{RMS}}
\] Here, \(K\) is a constant
determined by the BER specification of the system given in the following
Table
BER
Crest factor (K)
\(10^{-3}\)
6.18
\(10^{-4}\)
7.438
\(10^{-5}\)
8.53
\(10^{-6}\)
9.507
\(10^{-7}\)
10.399
\(10^{-8}\)
11.224
\(10^{-9}\)
11.996
\(10^{-10}\)
12.723
\(10^{-11}\)
13.412
\(10^{-12}\)
14.069
\(10^{-13}\)
14.698
1 2 3 4
K = 14.698; Ks = K/2; p = normcdf([-Ks Ks]); BER = 1 - (p(2)-p(1));
The phase noise is traditionally defined as the
ratio of the power of the signal in 1Hz bandwidth at offset \(f\) from the carrier \(P\), divided by the power of the carrier
\[
\ell (f) = \frac {S_v'(f_0+f)}{P}
\] where \(S_v'\) is is
one-sided voltage PSD and \(f
\geqslant 0\)
Under narrow angle assumption\[
S_{\varphi}(f)= \frac {S_v'(f_0+f)}{P}
\] where \(\forall f\in \left[-\infty
+\infty\right]\)
Using the Wiener-Khinchin theorem, it is possible to easily derive
the variance of the absolute jitter(\(J_{ee}\))via integration of the
corresponding PSD \[
J_{ee,rms}^2 = \int S_{J_{ee}}(f)df
\]
And we know the relationship between absolute jitter and excess phase
is \[
J_{ee}=\frac {\varphi}{\omega_0}
\] Considering that phase noise is normally symmetrical about the
zero frequency, multiplied by two is shown as below
\[
J_{ee,rms} = \frac{\sqrt{2\int_{0}^{+\infty}\ell(f)df}}{\omega_0}
\] where phase noise is in linear units not in logarithmic
ones.
Because the unit of phase noise in Spectre-RF is logarithmic
unit (dBc), we have to convert the unit before applying the
above equation \[
\ell[linear] = 10^{\frac {\ell [dBc/Hz]}{10}}
\] The complete equation using the simulation result of
Spectre-RF Pnoise is \[
J_{ee,rms} = \frac{\sqrt{2\int_{0}^{+\infty}10^{\frac {\ell
[dBc/Hz]}{10}}df}}{\omega_0}
\]
The above equation has been verified for sampled pnoise,
i.e. Jee and Edge Phase Noise.
For pnoise-sampled(jitter), Direct Plot Form -
Function: Jee:Integration Limits can calculate it conveniently
But for pnoise-timeaveage, you have to use the
below equation to get RMS jitter.
One example, integrate to \(\frac{f_{osc}}{2}\) and \(f_{osc} = 16GHz\)
Of course, it apply to conventional pnoise simulation.
On the other hand, output rms voltage noise, \(V_{out,rms}\) divied by slope should be
close to \(J_{ee,rms}\)\[
J_{ee,rms} = \frac {V_{out,rms}}{slope}
\]
Pnoise sampled: Edge Delay mode measures the noise
defined by two edges. Both edges are defined by a threshold voltage and
rising or falling edges, which measures the noise of the pulse itself
and direct plot calculate the variation of the pulse
width
Power supply induced jitter
(PSIJ)
A sampled pxf analysis can be used to simulate the
deterministic jitter of a circuit due to power supply
ripple
TODO 📅
DCC & AC-coupled buffer
The amount of correction can be set by intentional injection of an
offset current into the summing input node of INV,
threshold-adjustable inverter
Note that the change to the threshold is opposite in
direction to the change to INV
increasing DC of input signal is equivalent to lower down the
threshold of INV
Since duty-cycle error is high frequency component, the
high-pass filter suppresses the duty-cycle error propagating to the
output
The AC-coupling capacitor blocks the low-frequency component of the
input
The feedback resistor sets common mode voltage to the crossover
voltage
Bae, Woorham; Jeong, Deog-Kyoon: 'Analysis and Design of CMOS
Clocking Circuits for Low Phase Noise' (Materials, Circuits and Devices,
2020)
Casper B, O'Mahony F. Clocking analysis, implementation and
measurement techniques for high-speed data links: A tutorial. IEEE
Transactions on Circuits and Systems I: Regular Papers.
2009;56(1):17-39
reference
Article (20500632) Title: How to simulate Random and Deterministic
Jitters URL:
https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O3w000009fiXeEAI
J. Kim et al., "A 112 Gb/s PAM-4 56 Gb/s NRZ Reconfigurable
Transmitter With Three-Tap FFE in 10-nm FinFET," in IEEE Journal of
Solid-State Circuits, vol. 54, no. 1, pp. 29-42, Jan. 2019, doi:
10.1109/JSSC.2018.2874040
— et al., "A 224-Gb/s DAC-Based PAM-4 Quarter-Rate Transmitter With
8-Tap FFE in 10-nm FinFET," in IEEE Journal of Solid-State Circuits,
vol. 57, no. 1, pp. 6-20, Jan. 2022, doi: 10.1109/JSSC.2021.3108969
J. N. Tripathi, V. K. Sharma and H. Shrimali, "A Review on Power
Supply Induced Jitter," in IEEE Transactions on Components, Packaging
and Manufacturing Technology, vol. 9, no. 3, pp. 511-524, March 2019 [https://sci-hub.st/10.1109/TCPMT.2018.2872608]
H. Kim, J. Fan and C. Hwang, "Modeling of power supply induced jitter
(PSIJ) transfer function at inverter chains," 2017 IEEE International
Symposium on Electromagnetic Compatibility & Signal/Power Integrity
(EMCSI), Washington, DC, USA, 2017 [https://sci-hub.st/10.1109/ISEMC.2017.8077937]
Yin Sun, Chulsoon Hwang EMC Laboratory. Improving Power Supply
Induced Jitter Simulation Accuracy for IBIS Model [https://ibis.org/summits/aug20/sun.pdf]
Performing FFT to a signal with a large DC offset would
often result in a big impulse around frequency 0 Hz, thus masking out
the signals of interests with relatively small amplitude.
One method to remove DC offset from the original signal before
performing FFT
Subtracting the Mean of Original Signal
You can also not filter the input, but set zero to the zero frequency
point for FFT result.
Nyquist component
If we go back to the definition of the DFT \[
X(N/2)=\sum_{n=0}^{N-1}x[n]e^{-j2\pi
(N/2)n/2}=\sum_{n=0}^{N-1}x[n]e^{-j\pi n}=\sum_{n=0}^{N-1}x[n](-1)^n
\] which is a real number.
The discrete function \[
x[n]=\cos(\pi n)
\] is always \((-1)^n\) for
integer \(n\)
One general sinusoid at Nyquist and has phase shift \(\theta\), this is \(T=2\) and \(T_s=1\)
Moreover \(B \cdot (-1)^n = B\cdot \cos(\pi
n)\), then \[
B\cdot \cos(\pi n) = A \cdot \cos(\pi n + \theta)
\] We can NOT distinguish one from another.
Fs = 1000; % Sampling frequency T = 1/Fs; % Sampling period L = 1500; % Length of signal t = (0:L-1)*T; % Time vector S = 0.7*sin(2*pi*50*t) + sin(2*pi*120*t); X = S + 2*randn(size(t)); figure(1) plot(1000*t(1:50),X(1:50)) title('Signal Corrupted with Zero-Mean Random Noise') xlabel('t (milliseconds)') ylabel('X(t)')
figure(2) Y = fft(X); P2 = abs(Y/L); %!!! two-sided spectrum P2. P1 = P2(1:L/2+1); %!!! single-sided spectrum P1 P1(2:end-1) = 2*P1(2:end-1); % exclude DC and Nyquist freqency f = Fs*(0:(L/2))/L; figure(2) plot(f,P1) title('Single-Sided Amplitude Spectrum of X(t)') xlabel('f (Hz)') ylabel('|P1(f)|')
Alternative View
The direct current (DC) bin (\(k=0\)) and the bin at \(k=N/2\), i.e., the bin that corresponds to
the Nyquist frequency are purely real and unique.
sinusoidal waveform with \(10Hz\),
amplitude 1 is \(cos(2\pi f_c t)\). The
plot is shown as below with sampling frequency is \(20Hz\)
Amplitude and Phase spectrum, sampled with \(f_s=20\) Hz
The FFT magnitude of \(10Hz\) is
1 and its phase is 0 as shown as
above, which proves the DFT and IDFT.
Caution: the power of FFT is related to samples (DFT
Parseval's theorem), which may not be the power of continuous signal.
The average power of samples is ([1 -1 1 -1 -1 1 ...]) is
1, that of corresponding continuous signal is \(\frac{1}{2}\).
Power spectrum derived from FFT provide information of samples, i.e.
1
Moreover, average power of sample [1 -1 1 -1 1 ...] is same with DC
[1 1 1 1 ...].
A glitch is an unwanted pulse at the output of a
combinational logic network – a momentary change in an
output that should not have changed
A circuit with the potential for a glitch is said to have a
hazard
In other words a hazard is something intrinsic about a circuit; a
circuit with hazard may or may not have a glitch depending on input
patterns and the electric characteristics of the circuit.
When do circuits have hazards
?
Hazards are potential unwanted transients that occur in the output
when different paths from input to output have different propagation
delays
For use with --batch, perform automatic expansions as a
stand-alone tool. This sets up the appropriate Verilog mode environment,
updates automatics with M-x verilog-auto on all command-line files, and
saves the buffers. For proper results, multiple filenames need to be
passed on the command line in bottom-up order.
-f verilog-auto-save-compile
Update automatics with M-x verilog-auto, save the buffer, and
compile
Emacs
--no-site-file
Another file for site-customization is site-start.el.
Emacs loads this before the user's init file
(.emacs, .emacs.el or
.emacs.d/.emacs.d). You can inhibit the loading of this
file with the option --no-site-file
--batch
The command-line option --batch causes Emacs to run
noninteractively. The idea is that you specify Lisp programs to run;
when they are finished, Emacs should exit.
--load, -l FILE, load Emacs Lisp FILE using the load
function;
--funcall, -f FUNC, call Emacs Lisp function FUNC with
no arguments
-f FUNC
--funcall, -f FUNC, call Emacs Lisp function FUNC with
no arguments
--load, -l FILE
--load, -l FILE, load Emacs Lisp FILE using the load
function
Verilog-mode is a standard part of GNU Emacs as of 22.2.
multiple directories
AUTOINST only search in the file's directory
default.
You can append below verilog-library-directories for
multiple directories search
1 2 3
// Local Variables: // verilog-library-directories:("." "subdir" "subdir2") // End:
plusargs are command-line switches supported by the
simulator. As per SystemVerilog LRM arguments beginning with the
+ character will be available using the
$test$plusargs and $value$plusargsPLI
APIs.
1 2 3
$test$plusargs (user_string)
$value$plusargs (user_string, variable)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
// tb.v module tb; int a; initialbegin if($test$plusargs("RUNSIM")) begin $display("There is RUNSIM plusargs"); endelsebegin $display("There is NO $test$plusargs"); end if($value$plusargs("SEED=%d",a)) begin $display("SEED=%d",a); endelsebegin $display("There is NO $value$plusargs"); end end endmodule
compile
1 2
$ vlib work $ vlog -sv tb.v
simulate (QuestaSim)
without plusargs
1
$ vsim work.tb -c -do "run; exit"
1 2 3 4 5 6 7 8 9
# // # Loading sv_std.std # Loading work.tb(fast) # run # There is NO $test$plusargs # There is NO $value$plusargs # exit # End time: 13:04:23 on Jun 04,2022, Elapsed time: 0:00:01 # Errors: 0, Warnings: 0
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
reg in; ///////////////////////////////////////////////////// wire out; assign #2.5 out = in; ///////////////////////////////////////////////////// initialbegin in = 0; #16; in = 1; #2; in = 0; #10; in = 1; #4; in = 0; end
//////////// combination logic //////////////////////// always @(*) #2.5 out = in; /////////////////////////////////////////////////////// /* the above code is same with following code always @(*) begin #2.5; out = in; end */ initialbegin in = 0; #16; in = 1; #2; in = 0; #10; in = 1; #4; in = 0; end
alwaysbegin clk = 0; #5; foreverbegin clk = ~clk; #5; end end //////////// sequential logic ////////////////// always @(posedge clk) #2.5 out <= in; /////////////////////////////////////////////// initialbegin in = 0; #16; in = 1; #2; in = 0; #10; in = 1; end
initialbegin #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
reg in; reg out; /////////////// nonblocking assignment /// always @(*) begin out <= #2.5 in; end ///////////////////////////////////////// initialbegin in = 0; #16; in = 1; #2; in = 0; #10; in = 1; #4; in = 0; end
reg in; reg out; /////////////// blocking assignment /// always @(*) begin out = #2.5 in; end ///////////////////////////////////////// initialbegin in = 0; #16; in = 1; #2; in = 0; #10; in = 1; #4; in = 0; end
initialbegin #50; $finish(); end
endmodule
It seems that new event is discarded before previous
event is realized.
clocking block in
SystemVerilog
Assignment at <interface>.<clocking block>.<output
signal> (i.e. synchronous) do NOT change
<interface>.<output signal> until active clock edge.
What is the "+:" operator called in Verilog? [link]
A range of contiguous bits can be selected and is known as
part-select. There are two types of part-selects, one
with a constant part-select and another with an
indexed part-select
1 2
reg [31:0] addr; addr [23:16] = 8'h23; //bits 23 to 16 will be replaced by the new value 'h23 -> constant part-select
Having a variable part-select allows it to be used effectively in
loops to select parts of the vector. Although the starting bit can be
varied, the width has to be
constant.
[<start_bit +: ] // part-select increments from
start-bit
[<start_bit -: ] // part-select decrements from
start-bit
$display("unsigned out(%%d): %0d", outSumUs); $display("unsigned out(%%b): %b", outSumUs); end endmodule
xcelium
1 2 3 4 5 6 7
xcelium> run signed out(%d): -12 signed out(%b): 10100 unsigned out(%d): 20 unsigned out(%b): 10100 xmsim: *W,RNQUIE: Simulation is complete. xcelium> exit
vcs
1 2 3 4 5 6
Compiler version S-2021.09-SP2-1_Full64; Runtime version S-2021.09-SP2-1_Full64; May 7 17:24 2022 signed out(%d): -12 signed out(%b): 10100 unsigned out(%d): 20 unsigned out(%b): 10100 V C S S i m u l a t i o n R e p o r t
observation
When signed and unsigned is mixed,
the result is by default unsigned.
Prepend to operands with 0s instead of extending
sign, even though the operands is signed
LHS DONT affect how the simulator operate on the
operands but what the results represent, signed or unsigned
Therefore, although outSumUs is declared as signed, its
result is unsigned
subtraction example
In logic arithmetic, addition and subtraction are commonly used for
digital design. Subtraction is similar to addition except that the
subtracted number is 2's complement. By using 2's complement for the
subtracted number, both addition and subtraction can be unified to using
addition only.
$display("unsigned out(%%d): %0d", outSubUs); $display("unsigned out(%%b): %b", outSubUs); end endmodule
1 2 3 4 5 6
Compiler version S-2021.09-SP2-1_Full64; Runtime version S-2021.09-SP2-1_Full64; May 7 17:46 2022 signed out(%d): -3 signed out(%b): 11101 unsigned out(%d): 29 unsigned out(%b): 11101 V C S S i m u l a t i o n R e p o r t
1 2 3 4 5 6
xcelium> run signed out(%d): -3 signed out(%b): 11101 unsigned out(%d): 29 unsigned out(%b): 11101 xmsim: *W,RNQUIE: Simulation is complete.
$display("unsigned out(%%d): %0d", outSubUs); $display("unsigned out(%%b): %b", outSubUs); end endmodule
1 2 3 4 5 6
Compiler version S-2021.09-SP2-1_Full64; Runtime version S-2021.09-SP2-1_Full64; May 7 17:50 2022 signed out(%d): 13 signed out(%b): 01101 unsigned out(%d): 13 unsigned out(%b): 01101 V C S S i m u l a t i o n R e p o r t
1 2 3 4 5 6 7
xcelium> run signed out(%d): 13 signed out(%b): 01101 unsigned out(%d): 13 unsigned out(%b): 01101 xmsim: *W,RNQUIE: Simulation is complete. xcelium> exit
Verilog has a nasty habit of treating everything as unsigned unless
all variables in an expression are signed. To add insult to injury, most
tools won’t warn you if signed values are being ignored.
If you take one thing away from this post:
Never mix signed and unsigned variables in one
expression!
Chronologic VCS simulator copyright 1991-2021 Contains Synopsys proprietary information. Compiler version S-2021.09-SP2-2_Full64; Runtime version S-2021.09-SP2-2_Full64; Nov 19 11:02 2022 Coordinates (7,7): x : 00000111 7 y : 00000111 7 Move +4: x1: 00001011 11 *LOOKS OK* y1: 00001011 11 Move -4: x1: 00010011 19 *SURPRISE* y1: 00000011 3 V C S S i m u l a t i o n R e p o r t Time: 60 CPU Time: 0.260 seconds; Data structure size: 0.0Mb
// %t format will scale the rounded value to represent timeprecision // 1.1002*10e-9/1e-12 = 11002 $display("$realtime %%t = %t", $realtime); $display("$time %%t = %t", $time); end endmodule
The time unit and time precision can be specified in the following
two ways:
Using the compiler directive `timescale
Using the keywords timeunit and
timeprecision
1 2 3 4 5 6 7 8 9 10
module D (...); timeunit100ps; timeprecision10fs; ... endmodule
module E (...); timeunit100ps / 10fs; // timeunit with optional second argument ... endmodule
The minimum of timeprecision determine %t
output, the nearest timeunit and timeprecision
determine the round of $realtime and $time. Of
course, the simulator follow the time tick shown by
$realtime.
It specifies the FSDB file name created by the Novas object files for
FSDB dumping. If it is not specified, then the default FSDB file name is
"novas.fsdb".
This command is valid only before executing
$fsdbDumpvars and is ignored if specified after
$fsdbDumpvars
$fsdbSuppress
The fsdbSuppressutility is used to skip dumping of few
instances, scopes, modules and signals. The
fsdbSuppressutility is a system task like other fsdb
tasks.
For $fsdbSuppress() to be effective, it needs to be
specified/called before$fsdbDumpvars
$fsdbAutoSwitchDumpfile
Automatically switch to a new dump file when the working FSDB file
reaches the specified size or the specified wall time period.
After the dumping is finished, a virtual FSDB file
(*.vf) is automatically created and list all of the generated
FSDB files with the correct sequence. Only the virtual FSDB
file, rather than all of the FSDB files, needs to be loaded to
view the simulation results
When specified in the design to switch based on file
size:
For VCS users, to include memory, MDA, packed array and structure
information in the generated FSDB file, the -debug_access
option must be included when VCS is invoked to compile the
design
depth
Specify how many sub-scope levels under the given scope you want to
dump.
Specify this argument as 1 to dump the signals
under the given scope
Specify this argument as 0 to dump all signals
under the given scope and its descendant scopes.
0: all signals in all scopes.
1: all signals in current scope.
2: all signals in the current scope and all scopes one level
below.
n: all signals in the current scope and all scopes n-1 levels
below.
initialbegin clk = 1'b0; forever #5 clk = ~clk; end
initialbegin #100; $finish(); end
initialbegin #10; $fsdbDumpfile("tb.fsdb"); //$fsdbDumpvars(0); // same with $fsdbDumpvars(0, tb) //$fsdbDumpvars(1); // same with $fsdbDumpvars(1, tb) //$fsdbDumpvars(2); // same with $fsdbDumpvars(2, tb) //$fsdbDumpvars(1, tb.u_div2); $fsdbDumpvars(0, tb.u_div2); #80$finish(); end
endmodule
module divider2 ( input clk ); reg div2;
divider2neg u_div2neg(div2); always@(posedge clk) begin div2 = ~div2; end initialbegin div2 = 1'b0; end
endmodule
module divider2neg ( input clk ); reg div2neg;
always@(negedge clk) begin div2neg = ~div2neg; end initialbegin div2neg= 1'b0; end
endmodule
compile
1
vcs -full64 -kdb -debug_access+all tb.v
simulate
1
./simv
load fsdb
1
verdi -ssf tb.fsdb
$fsdbDumpon, $fsdbDumpoff
1 2 3
$fsdbDumpon(["+fsdbfile+filename"])
$fsdbDumpoff(["+fsdbfile+filename"])
These FSDB dumping commands turn dumping on and off.
fsdbDumpon/fsdbDumpoff has the highest priority and
overrides all other FSDB dumping commands.
fsdbDumpon/fsdbDumpoff is not restricted to only
fsdbDumpvars. If there is more than one FSDB file open for
dumping at one simulation run, fsdbDumpon/fsdbDumpoff may
only affect a specific FSDB file by specifying the specific file
name.
+fsdbfile+filename: Specify the FSDB file name. If not
specified, the default FSDB file name is "novas.fsdb"
$fsdbDumpFinish
This command closes all FSDB files in the current simulation and
stops dumping of signals. Although all FSDB files are closed
automatically at the end of simulation, this dumping command can be
invoked to explicitly close the FSDB files during the
simulation
VCD
$dumpfile
The declaration onf $dumpfile must come before the
$dumpvars or any other system tasks that specifies
dump.
1
$dumpfile("test.vcd");
argument is necessary, there is no default value
$dumpvars
The $dumpvars is used to specify which variables are to
be dumped ( in the file mentioned by $dumpfile). The
simplest way to use it is without any argument.
1
$dumpvars(<levels> <, <module_or_variable>>* );
$dumplimit
It is possible that you inadvertantly generate huge file in Gigabytes
( for examples while dumping a Gigahertz clock for one second). To
reduce such occurrences, we may use $dumplimit. It usage
is
1
$dumplimit(<filesize>);
$dumpoff and $dumpon
During the simulation if you are bothered about about only during a
certain interval then you can use $dumpoff and
$dumpon. The following example shows its usage. It will
dump the changes for first 100 units of time and then between 10200 and
10400 units of time.
The system task has no versions to accept octal data or decimal
data.
The 1st argument is the data file name.
The 2nd argument is the array to receive the data.
The 3rd argument is an optional start address, and if you provide
it, you can also provide
The 4th argument optional end address.
Note, the 3rd and 4th argument address is for array not data
file.
If the memory addresses are not specified anywhere, then the system
tasks load file data sequentially from the lowest address toward the
highest address.
The standard before 2005 specify that the system tasks load file data
sequentially from the left memory address bound to the right memory
address bound.
With implict sign extension, the implementation of
signed arithmetic is DIFFERENT from
that of unsigned. Otherwise, their implementations are
same.
The implementations manifest the RTL's behaviour correctly
add without implicit sign
extension
unsigned
rtl
1 2 3 4 5 6 7
module TOP ( inputwire [2:0] data0 ,inputwire [2:0] data1 ,outputwire [2:0] result ); assign result = data0 + data1; endmodule
synthesized netlist
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
///////////////////////////////////////////////////////////// // Created by: Synopsys DC Ultra(TM) in wire load mode // Version : S-2021.06-SP5 // Date : Sat May 7 11:43:27 2022 /////////////////////////////////////////////////////////////
module TOP ( data0, data1, result ); input [2:0] data0; input [2:0] data1; output [2:0] result; wire n4, n5, n6;
module TOP ( inputwiresigned [2:0] data0 ,inputwiresigned [2:0] data1 ,outputwiresigned [2:0] result ); assign result = data0 + data1; endmodule
synthesized netlist
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
///////////////////////////////////////////////////////////// // Created by: Synopsys DC Ultra(TM) in wire load mode // Version : S-2021.06-SP5 // Date : Sat May 7 11:48:54 2022 /////////////////////////////////////////////////////////////
module TOP ( data0, data1, result ); input [2:0] data0; input [2:0] data1; output [2:0] result; wire n4, n5, n6;
module TOP ( inputwire [2:0] data0 // 3 bit unsigned ,inputwire [1:0] data1 // 2 bit unsigned ,outputwire [2:0] result // 3 bit unsigned ); assign result = data0 + data1; endmodule
synthesized netlist
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
///////////////////////////////////////////////////////////// // Created by: Synopsys DC Ultra(TM) in wire load mode // Version : S-2021.06-SP5 // Date : Sat May 7 12:15:58 2022 /////////////////////////////////////////////////////////////
module TOP ( data0, data1, result ); input [2:0] data0; input [1:0] data1; output [2:0] result; wire n4, n5, n6;
///////////////////////////////////////////////////////////// // Created by: Synopsys DC Ultra(TM) in wire load mode // Version : S-2021.06-SP5 // Date : Sat May 7 12:21:51 2022 /////////////////////////////////////////////////////////////
UC Berkeley CS150 Lec #20: Finite State Machines [slides]
always@( * )
always@( * ) blocks are used to describe Combinational
Logic, or Logic Gates. Only = (blocking) assignments should
be used in an always@( * ) block.
Latch Inference
If you DON'T assign every element that can be
assigned inside an always@( * ) block every time that
always@( * ) block is executed, a latch will be inferred
for that element
The approaches to avoid latch generation:
set default values
proper use of the else statement, and other flow
constructs
without default values
latch is generated
RTL
1 2 3 4 5 6 7 8 9 10 11 12 13 14
module TOP ( inputwire Trigger, inputwire Pass, outputreg A, outputreg C ); always @(*) begin A = 1'b0; if (Trigger) begin A = Pass; C = Pass; end end endmodule
synthesized netlist
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
///////////////////////////////////////////////////////////// // Created by: Synopsys DC Ultra(TM) in wire load mode // Version : S-2021.06-SP5 // Date : Mon May 9 17:09:18 2022 /////////////////////////////////////////////////////////////
module TOP ( Trigger, Pass, A, C ); input Trigger, Pass; output A, C;
Default values are an easy way to avoid latch generation
RTL
1 2 3 4 5 6 7 8 9 10 11 12 13 14
module TOP ( inputwire Trigger, inputwire Pass, outputreg A, outputreg C ); always @(*) begin A = 1'b0; C = 1'b1; if (Trigger) begin A = Pass; C = Pass; end end
synthesized netlist
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
///////////////////////////////////////////////////////////// // Created by: Synopsys DC Ultra(TM) in wire load mode // Version : S-2021.06-SP5 // Date : Mon May 9 17:12:47 2022 /////////////////////////////////////////////////////////////
module TOP ( Trigger, Pass, A, C ); input Trigger, Pass; output A, C;
regsigned [8:0] t; // extend 1b begin if(plus) begin t = {a[7], a} + {1'b0, b}; satop_sus8b = (t[8:7]==2'b01) ? {1'b0, {7{1'b1}}} // up saturate for signed : t[7:0]; endelsebegin t = {a[7], a} - {1'b0, b}; satop_sus8b = (t[8:7]==2'b10) ? {1'b1, {7{1'b0}}} // dn saturate for signed : t[7:0]; end end endfunction
A power spectrum is equal to the square of the absolute value of
DFT.
The sum of all power spectral lines in a power spectrum is equal to
the power of the input signal.
The integral of a PSD is equal to the power of the input
signal.
power spectrum has units of \(V^2\)
and power spectral density has units of \(V^2/Hz\)
Parseval's theorem is a property of the Discrete
Fourier Transform (DFT) that states: \[
\sum_{n=0}^{N-1}|x(n)|^2 = \frac{1}{N}\sum_{k=0}^{N-1}|X(k)|^2
\] Multiply both sides of the above by \(1/N\): \[
\frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2 =
\frac{1}{N^2}\sum_{k=0}^{N-1}|X(k)|^2
\]\(|x(n)|^2\) is instantaneous
power of a sample of the time signal. So the left side of the equation
is just the average power of the signal over the N
samples. \[
P_{\text{av}} = \frac{1}{N^2}\sum_{k=0}^{N-1}|X(k)|^2\text{, }V^2
\] For the each DFT bin, we can say: \[
P_{\text{bin}}(k) = \frac{1}{N^2}|X(k)|^2\text{,
k=0:N-1, }V^2/\text{bin}
\] This is the power spectrum of the signal.
Note that \(X(k)\) is the
two-sided spectrum. If \(x(n)\) is real, then \(X(k)\) is symmetric about \(fs/2\), with each side containing half of
the power. In that case, we can choose to keep just the
one-sided spectrum, and multiply Pbin by 2
(except DC & Nyquist):
rng default Fs = 1000; t = 0:1/Fs:1-1/Fs; x = cos(2*pi*100*t) + randn(size(t)); N = length(x); xdft = fft(x); xsq_sum_avg = sum(x.^2)/N; specsq_sum_avg = sum(abs(xdft).^2)/N^2;
where xsq_sum_avg is same with
specsq_sum_avg
For a discrete-time sequence x(n), the DFT is defined as: \[
X(k) = \sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N}
\] By it definition, the DFT does NOT apply to
infinite duration signals.
Different scaling is needed to apply for amplitude spectrum, power
spectrum and power spectrum density, which shown as below
\(f_s\) in Eq.(13) is sample
rate rather than frequency resolution.
And Eq.(13) can be expressed as \[
\text{PSD}(k) =\frac{1}{f_{\text{res}}\cdot
N\sum_{n}w^2(n)}\left|X_{\omega}(k)\right|^2
\] where \(f_{\text{res}}\) is
frequency resolution
We define the following two sums for normalization purposes:
where Normalized Equivalent Noise BandWidth is
defined as \[
\text{NENBW} =\frac{N S_2}{S_1^2}
\] and Effective Noise BandWidth is \[
\text{ENBW} =f_{\text{res}} \cdot \frac{N S_2}{S_1^2}
\]
For Rectangular window, \(\text{ENBW}
=f_{\text{res}}\)
This equivalent noise bandwidth is required when the
resulting spectrum is to be expressed as spectral density (such as
for noise measurements).
plot(Fx1,10*log10(Pxx1),Fx2,10*log10(Pxx2),'r--'); legend('PSD via Eq.(13)','PSD via pwelch')
window effects
It is possible to correct both the amplitude and energy content of
the windowed signal to equal the original signal. However, both
corrections cannot be applied simultaneously
power spectral density (PSD)\[
\text{PSD} =\frac{\left|X_{\omega}(k)\right|^2}{f_s\cdot S_2}
\]
We have \(\text{PSD} =
\frac{\text{PS}}{\text{ENBW}}\), where \(\text{ENBW}=\frac{N \cdot
S_2}{S_1^2}f_{\text{res}}\)
linear power spectrum\[
\text{PS}_L=\frac{|X_{\omega}(k)|^2}{N\cdot S_2}
\]
usage: RMS value, total power \[
\text{PS}_L(k)=\text{PSD(k)} \cdot f_{\text{res}}
\]
Window Correction Factors
While a window helps reduce leakage (The window reduces the jumps at
the ends of the repeated signal), the window itself distorts the data in
two different ways:
Amplitude – The amplitude of the signal
is reduced
This is due to the fact that the window removes information in the
signal
Energy – The area under the curve, or
energy of the signal, is reduced
Window correction factors are used to try and
compensate for the effects of applying a window to data. There are both
amplitude and energy correction factors.
Window Type
Amplitude Correction (\(K_a\))
Energy Correction (\(K_e\))
Rectangluar
1.0
1.0
hann
1.9922
1.6298
blackman
2.3903
1.8155
kaiser
1.0206
1.0204
Only the Uniform window (rectangular window), which is equivalent to
no window, has the same amplitude and energy correction factors.
In literature, Coherent power gain is defined show
below, which is close related to \(K_a\)\[
\text{Coherent power gain (dB)} = 20 \; log_{10} \left( \frac{\sum_n
w[n]}{N} \right)
\]
With amplitude correction, by multiplying by two, the peak
value of both the original and corrected spectrum match. However
the energy content is not the same.
The amplitude corrected signal (red) appears to have more energy, or
area under the curve, than the original signal (blue).
Multiplying the values in the spectrum by 1.63, rather than 2, makes
the area under the curve the same for both the original signal
(blue) and energy corrected signal (red)
hanning's correction factors:
1 2 3 4
N = 256; w = hanning(N); Ka = N/sum(w) Ke = sqrt(N/sum(w.^2))
%% plot psd of two methods plot(Fx1,10*log10(Pxx1),Fx2,10*log10(Pxx2),'r--'); legend('PSD via Eq.(13)','PSD via pwelch') grid on; xlabel('Freq (Hz)'); ylabel('dB; V^2/Hz')
We may also want to know for example the RMS value of the signal, in
order to know how much power the signal generates. This can be done
using Parseval’s theorem.
For a periodic signal, which has a discrete
spectrum, we obtain its total RMS value by summing the included signals
using \[
x_{\text{rms}}=\sqrt{\sum R_{xk}^2}
\] Where \(R_{xk}\) is the RMS
value of each sinusoid for \(k=1,2,3,...\) The RMS value of a signal
consisting of a number of sinusoids is consequently equal to the
square root of the sum of the RMS values.
This result could also be explained by noting that sinusoids of
different frequencies are orthogonal, and can therefore
be summed like vectors (using Pythagoras’ theorem)
For a random signal we cannot interpret the spectrum in the same way.
As we have stated earlier, the PSD of a random signal contains all
frequencies in a particular frequency band, which makes it
impossible to add the frequencies up. Instead, as the PSD is a
density function, the correct interpretation is to sum the area under
the PSD in a specific frequency range, which then is the square of the
RMS, i.e., the mean-square value of the signal \[
x_{\text{rms}}=\sqrt{\int G_{xx}(f)df}=\sqrt{\text{area under the
curve}}
\] The linear spectrum, or RMS
spectrum, defined by \[\begin{align}
X_L(k) &= \sqrt{\text{PSD(k)} \cdot f_{\text{res}}}\\
&=\sqrt{\frac{\left|X_{\omega}(k)\right|^2}{f_{\text{res}}\cdot
N\sum_{n}\omega^2(n)} \cdot f_{\text{res}}} \\
&= \sqrt{\frac{\left|X_{\omega}(k)\right|^2}{N\sum_{n}\omega^2(n)}}
\\
&= \sqrt{\frac{|X_{\omega}(k)|^2}{N\cdot S_2}}
\end{align}\]
The corresponding linear power spectrum or
RMS power spectrum can be defined by \[\begin{align}
\text{PS}_L(k)&=X_L(k)^2=\frac{|X_{\omega}(k)|^2}{S_1^2}\frac{S_1^2}{N\cdot
S_2} \\
&=\text{PS}(k) \cdot \frac{S_1^2}{N\cdot S_2}
\end{align}\]
So, RMS can be calculated as below \[\begin{align}
P_{\text{tot}} &= \sum \text{PS}_L(k) \\
\text{RMS} &= \sqrt{P_{\text{tot}}}
\end{align}\]
DFT averaging
we use \(N= 8*\text{nfft}\) time
samples of \(x\) and set the number of
overlapping samples to \(\text{nfft}/2 =
512\). pwelch takes the DFT of \(\text{Navg} = 15\) overlapping segments of
\(x\), each of length \(\text{nfft}\), then averages the \(|X(k)|^2\) of the DFT’s.
In general, if there are an integer number of segments that cover all
samples of N, we have \[
N = (N_{\text{avg}}-1)*D + \text{nfft}
\] where \(D=\text{nfft}-\text{noverlap}\). For our
case, with \(D = \text{nfft}/2\) and
\(N/\text{nfft} = 8\), we have \[
N_{\text{avg}}=2*N/\text{nfft}-1=15
\] For a given number of time samples N, using overlapping
segments lets us increase \(N_{avg}\)
compared with no overlapping. In this case, overlapping of 50% increases
\(N_{avg}\) from 8 to 15. Here is the
Matlab code to compute the spectrum:
1 2 3 4 5 6 7 8
nfft= 1024; N= nfft*8; % number of samples in signal n= 0:N-1; x= A*sin(2*pi*f0*n*Ts) + .1*randn(1,N); % 1 W sinewave + noise noverlap= nfft/2; % number of overlapping time samples window= hanning(nfft); [pxx,f]= pwelch(x,window,noverlap,nfft,fs); % W/Hz PSD PdB_bin= 10*log10(pxx*fs/nfft); % dBW/bin
DFT averaging reduces the variance \(\sigma^2\) of the noise spectrum by a
factor of \(N_{avg}\), as long as
noverlap is not greater than nfft/2
fftshift
The result of fft and its index is shown as below
After fftshift
1 2 3 4 5
>> fftshift([01234567])
ans =
45670123
1 2 3 4 5 6 7 8 9
clear; N = 100; fs = 1000; fx = 100; x = cos(2*pi*fx/fs*[0:1:N-1]); s = abs(fftshift(fft(x)))/N; fx = linspace(0,N-1,N)*fs/N -fs/2; plot(fx, s, 'o-', 'linewidth', 2); grid on; grid minor;
dft and
psd function in virtuoso
dft always return
To compensate windowing effect, \(W(n)\), the dft output should
be multiplied by \(K_a\), e.g. 1.9922
for hanning window.
psd function has taken into account \(K_e\), postprocessing is
not needed
reference
Heinzel, Gerhard, A. O. Rüdiger and Roland Schilling. "Spectrum and
spectral density estimation by the Discrete Fourier transform (DFT),
including a comprehensive list of window functions and some new at-top
windows." (2002). URL: https://holometer.fnal.gov/GH_FFT.pdf
Rapuano, Sergio, and Harris, Fred J., An Introduction to FFT and Time
Domain Windows, IEEE Instrumentation and Measurement Magazine, December,
2007. https://ieeexplore.ieee.org/document/4428580
Jens Ahrens, Carl Andersson, Patrik Höstmad, Wolfgang Kropp,
“Tutorial on Scaling of the Discrete Fourier Transform and the Implied
Physical Units of the Spectra of Time-Discrete Signals” in 148th
Convention of the AES, e-Brief 56, May 2020 [ pdf,
web
].
Manolakis, D., & Ingle, V. (2011). Applied Digital Signal
Processing: Theory and Practice. Cambridge: Cambridge University
Press. doi:10.1017/CBO9780511835261
In T* DRC deck, it is based on the voltage recognition CAD layer and
net connection to calculate the voltage difference between two
neighboring nets by the following formula:
\[
\Delta V = \max(V_H(\text{net1})-V_L(\text{net2}),
V_H(\text{net2})-V_L(\text{net1}))
\]
where \[
V_H(\text{netx}) = \max(V(\text{netx}))
\] and \[
V_L(\text{netx}) = \min(V(\text{netx}))
\]
The \(\Delta V\) will be
0 if two nets are connected as same potential
If \(V_L \gt V_H\)on a
net, DRC will report warning on this net
Voltage recognition CAD
Layer
Two method
voltage text layer
You place specific voltage text on specific drawing layer
voltage marker layer
Each voltage marker layer represent different voltage for specific
drawing layer
voltage text layer has higher priority than voltage
marker layer and is recommended
voltage text layer
For example M3
Process Layer
CAD Layer#
Voltage High
Voltage High Top (highest priority)
Voltage Low
Voltage Low Top (highest priority)
M3
63
110
112
111
113
where 63 is layer number,
110 ~ 113 is datatype
voltage marker layer
Different data type represent different voltage, like
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\(\frac{d}{dt}x(t)\)
Laplace Transform
Bilateral Laplace Transform
\(sX(s)\)
Unilateral Laplace Transform
\(sX(s)-x(0^-)\)
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
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 \[
\lim_{n\to\infty}f[n]=\lim_{z\to1}(z-1)F(z)
\]
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
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
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 equivalenceonly 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,
%% 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(:));
%% 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
\[\begin{align}
P_{\text{xx}}'(\omega) &= P_{\text{xx}}(\omega) \cdot \left|
1-z^{-k} \right|^2 \\
&= P_{\text{xx}}(\omega) \cdot \left| 1-(e^{j\omega
T_s})^{-k} \right|^2 \\
&= P_{\text{xx}}(\omega) \cdot \left| 1-e^{-j\omega T_s
k} \right|^2
\end{align}\]
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 invariancecannot 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
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
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
\[
s \approx \frac{1-z^{-1}}{T}
\]
Suppose we wish to make a discrete-time filter based on a prototype
first-order low-pass filter \[
H_p(s) = \frac{1}{s\tau + 1}
\] The differential equation describing this filter is \[
\tau\frac{dy}{dt} + y = x
\] then differential equation gives
\[
\tau\frac{y_n-y_{n-1}}{T} + y_n=x_n
\] or \[
(\frac{\tau}{T}+1)y_n - \frac{\tau}{T}y_{n-1} = x_n
\] The transfer function is \[
H(z) = \frac{\frac{T}{T+\tau}}{1-\frac{\tau}{T+\tau}z^{-1}}
\]
or substitute \(s\) with \(\frac{1-z^{-1}}{T}\) into \(H_p(s)\)\[
H_p(z) = \frac{1}{\tau\frac{1-z^{-1}}{T} + 1} =
\frac{\frac{T}{T+\tau}}{1-\frac{\tau}{T+\tau}z^{-1}}= \frac{\alpha}{1
+(\alpha -1)z^{-1}}
\] where \(\alpha =
\frac{T}{\tau+T}\)
Under the assumption, time constants much longer than the timestep
\(\tau \gg T\)\[\begin{align}
\frac{T}{T+\tau}& = \frac{T}{\tau}\cdot \frac{\tau}{T+\tau}\approx
\frac{T}{\tau} \\
\frac{\tau}{T+\tau} &= \frac{\tau -
T}{\tau}\cdot\frac{\tau^2}{\tau^2-T^2} \approx \frac{\tau - T}{\tau} =
1- \frac{T}{\tau}
\end{align}\]
Then the first-order low pass fiter \[
H_p(z) \approx \frac{\alpha}{1 +(\alpha -1)z^{-1}}
\] where \(\alpha =
\frac{T}{\tau}\)
1 2 3 4 5 6 7 8 9 10 11 12 13
# https://www.dsprelated.com/showarticle/1517/return-of-the-delta-sigma-modulators-part-1-modulation # https://www.embeddedrelated.com/showarticle/779.php defshow_dsmod_samplewave(t,x,dsmod,args=(1,),tau=0.05, R=1, fig=None, return_handles=False, filter_dsmod=False): dt = t[1]-t[0] if filter_dsmod: x1 = dsmod(x, *args,R=R, modulate=False) else: x1 = x y = dsmod(x,*args,R=R) a = dt/tau yfilt = scipy.signal.lfilter([a],[1,a-1],y) xfilt = scipy.signal.lfilter([a],[1,a-1],x1)
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
\[
z = e^{sT}
\]
1 2 3 4 5 6 7 8 9 10
% 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)
% 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= [1221]; [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')
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 \[
H(\omega) = \frac{Y_{\text{pulse}}(\omega)}{X_{\text{rect}}(\omega)} =
\frac{Y_{\text{pulse}}(\omega)}{\text{sinc}(\omega)}
\]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
% 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);
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.
\[
f_\text{SRF} = \frac{1}{2\pi \sqrt{LC}}
\] 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
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.
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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