KNOW-HOW

unsigned + unsigned = unsigned

1
2
3
4
5
6
7
8
9
10

function [7:0] satadd_uuu8b;   // unsigned + unsigned = unsigned
    input [7:0] a;
    input [7:0] b;

    reg [8:0] t;   // extend 1b
    begin
        t = {1'b0, a} + {1'b0, b};
        satop_uuu16b = t[8] ? {8{1'b1}} : t[7:0];
    end
endfunction

1'b1: overflow

signed + signed = signed

1
2
3
4
5
6
7
8
9
10
11
12

function [7:0] satadd_sss8b;    // signed + signed = signed
    input signed    [7:0] a;
    input signed    [7:0] b;

    reg signed [8:0] t; // extend 1b
    begin
        t = a + b;  // extend sign bit automatically
        satadd_sss8b =  (t[8:7] == 2'b01) ? {1'b0, 7{1'b1}} : // up sat
                        (t[8:7] == 2'b10) ? {1'b1, 7{1'b0}} : // dn sat
                                            t[7:0];
    end
endfunction

2'b01: overflow

2'b10: underflow

signed + unsigned = unsigned

1
2
3
4
5
6
7
8
9
10
11
12

function [7:0] satadd_suu8b;    // signed + unsigned = unsigned
    input signed    [7:0] a;
    input           [7:0] b;

    reg signed [8:0] t; // extend 1b
    begin
        t = {a[7], a} + {1'b0, b};
        satadd_ssu8b =  (t[8:7] == 2'b10) ? {8{1'b1}} : // up saturate for unsigned
                        (t[8:7] == 2'b11) ? {8{1'b0}} : // dn saturate for unsigned
                                            t[7:0];
    end
endfunction

signed + unsigned = signed

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

function signed [7:0] satop_sus8b;    //signed +/- unsigned = signed
    input signed    [7:0] a;
    input           [7:0] b;
    input plus;

    reg signed [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];
        end else begin
            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

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

Always-On Buffer

reference

Isolation cells and Level Shifter cells URL: https://vlsitutorials.com/isolation-cells-level-shifter-cells-low-power-vlsi/

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

Jitter Components

dual-Dirac model

Spectral method

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.

余宥浚 Jacky Yu, Keysight Taiwan AEO, Advanced Jitter and Eye-Diagram Analysis

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's phd thesis URL: https://dr.lib.iastate.edu/handle/20.500.12876/30459

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., 2006, pp. 6 pp.-, doi: 10.1109/ASPDAC.2006.1594696.

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.

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);

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.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52

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

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

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

Deterministic Jitter

j_Djpp can be calculated by PSD,too

1
2
3
4
5
6
7
8
9

fck = 38.4e6;
Nfft = 15000;
fres = fck/Nfft;
psddBc = -99.3343;
psBc = psddBc + 10*log10(fres);	% psd -> ps; 
phrad2 = 10^(psBc/10);
phrms = sqrt(phrad2);
Jrms = phrms/2/pi*1/fck;
Jpp = 2*sqrt(2)*Jrms;

1
2
3

Jpp =

   6.4038e-12

For DJ, we usually use peak to peak value

BTW, the psd value at half of fundamental frequency (\(f_s/2\)) is misleading and ambiguity, we won't use this value

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

1
2
3
4

K = 14.698;
Ks = K/2;
p = normcdf([-Ks Ks]);
BER = 1 - (p(2)-p(1));

1
2
3

BER =

   1.9962e-13

Total Jitter

\[ \text{TJ}_{\text{p-p}}\equiv \text{DJ}_{\text{p-p}} + \text{RJ}_{\text{p-p}}(\text{BER}) \]

In the psd of TJ, the spur is DJ and floor is RJ

Phase Noise to Jitter

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. J_ee 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 \(J_{ee,rms}\) \[ J_{ee,rms} = \frac {V_{out,rms}}{slope} \]

reference

Article (20500632) Title: How to simulate Random and Deterministic Jitters URL: https://support.cadence.com/apex/ArticleAttachmentPortal?id=a1O3w000009fiXeEAI

Spectre Tech Tips: Measuring Noise in Digital Circuits - Analog/Custom Design - Cadence Blogs - Cadence Community https://community.cadence.com/cadence_blogs_8/b/cic/posts/s . . .

DC offset

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 \(T_s=1\)

\[\begin{align} x[n] &= A \cos(\pi n + \theta) \\ &= A \big( \cos(\pi n) \cos(\theta) - \sin(\pi n) \sin(\theta) \big) \\ &= \big(A\cos(\theta)\big) \cos(\pi n) + \big(-A\sin(\theta)\big) \sin(\pi n) \\ &= \big(A\cos(\theta)\big) (-1)^n + \big(-A\sin(\theta)\big) \cdot 0 \\ &= B \cdot (-1)^n \\ \end{align}\]

Where \(A\cos(\theta)=B\).

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.

In other words, you CAN'T infer the signal from \(X(\frac{N}{2})\) \[\begin{align} X(k)\frac{1}{N}e^{j 2 \pi \frac{nk}{N}}\bigg|_{k=\frac{N}{2}} &= \frac{X\left(\frac{N}{2} \right)}{N}(-1)^n \\ &= \frac{X\left(\frac{N}{2} \right)}{N}\cos(\pi n) \\ &= \frac{X\left(\frac{N}{2} \right)}{N}\left( \cos(\pi n) - \beta \sin(\pi n) \right) \\ &= \frac{X\left(\frac{N}{2} \right)}{N}\sqrt{1+\beta^2}\left(\frac{1}{\sqrt{1+\beta^2}} \cos(\pi n) - \frac{\beta}{\sqrt{1+\beta^2}} \sin(\pi n) \right) \\ &= \frac{X\left(\frac{N}{2} \right)}{N} \frac{1}{\cos(\theta)}\left(\cos(\theta) \cos(\pi n) - \sin(\theta) \sin(\pi n) \right) \\ &= \frac{X\left(\frac{N}{2} \right)}{N} \frac{1}{\cos(\theta)} \cos(\pi n+\theta) \end{align}\]

where \(\beta \in \mathbb{R}\) and you wouldn't know it because \(\sin(\pi n)=0 \quad \forall n \in \mathbb{Z}\)

For example, if \(\theta=0\) \[ X(k)\frac{1}{N}e^{j 2 \pi \frac{nk}{N}}\bigg|_{k=\frac{N}{2}}= \frac{X\left(\frac{N}{2} \right)}{N} \cos(\pi n) \] However, if \(\theta=\frac{\pi}{3}\) \[ X(k)\frac{1}{N}e^{j 2 \pi \frac{nk}{N}}\bigg|_{k=\frac{N}{2}}= \frac{X\left(\frac{N}{2} \right)}{N}\cdot 2 \cos(\pi n+\frac{\pi}{3}) \]

That sort of ambiguity is the reason for the strict inequality of the sampling theorem's condition.

Duty Cycle Distortion

Both edges is used for clock waveform to evaluate the duty cycle distortion,

Assuming TIE is [0 0.2 0 0.2 0 0.2 ...], then subtract DC offset, we get [-0.1 0.1 -0.1 0.1 ...], shown as below

The amplitude is manifested in FFT amplitude spectrum, i.e. Nyquist component, which is 0.1 in follow figure

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44

N = 32;
n = (1:N);
x = 0.1*(-1).^n;
figure(1)
stem(n-1, x);
X = fft(x)/N;
Xshift = fftshift(X);
fprintf("nyquist component: %.2f\n", Xshift(1));
magXshift = abs(Xshift);
ph = phase(Xshift)/pi*180;
figure(2)
subplot(3, 1, 1)
fx = (-N/2:N/2-1);
stem(fx, magXshift);
xlabel('Freq');
ylabel('|X(k)|');
title('mag of DFT');
grid on;

subplot(3, 1, 2)
stem(fx, ph);
xlabel('Freq');
ylabel('\angle X(k)(^oC)');
title('phase of DFT');
grid on;

%% inverse dft
ninv = (0:32-1);
xinv = Xshift(1)*cos(pi*ninv);
subplot(3, 1, 3);
hold on;
stem(ninv, x, "filled", 'r');
stem(ninv, xinv,'bd-.');
ninfer = (0:0.1:32+1);
xinfer1 = Xshift(1)*cos(pi*ninfer); % theta = 0
xinfer2 = Xshift(1)*2*cos(pi*ninfer+pi/3); % theta = pi/3
plot(ninfer, xinfer1, 'm--');
plot(ninfer, xinfer2, 'c--');
hold off;
legend('original', 'IDFT', '\theta=0', '\theta=\pi/3');
xlabel('time');
ylabel('V');
title('sample points');
grid on;

Single-Sided Amplitude Spectrum

DC and Nyquist frequency of FFT left over

P1(2:end-1) = 2*P1(2:end-1);

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

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 ...].

code

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42

fc = 10;
fs = 2*fc;
fov = 64*fc;

ts = (0:1/fs:26);
tov = (0:1/fov:26);

ys = cos(2*pi*fc*ts);
yov = cos(2*pi*fc*tov);
%% waveform
stem(ts, ys)
hold on;
plot(tov, yov);
legend('sample', 'waveform')
ylim([-1.5 1.5])
grid on;
xlabel('time(s)');
ylabel('mag(V)')

nfft = 256;
X = fftshift(fft(ys, nfft))/nfft;
f = (-nfft/2:nfft/2-1)*fs/nfft;
magX = abs(X);
phsX = atan2(imag(X),real(X));
%% fft spectrum
figure(2)
subplot(2, 1, 1);
stem(f, magX);
xlabel('Frequency(Hz)');
ylabel('mag')
xlim([min(f)-1 max(f)+1])
title('Amplitude spectrum')
subplot(2, 1, 2)
plot(f, phsX);
xlabel('Frequency(Hz)');
ylabel('phase (rad)')
xlim([min(f)-1 max(f)+1])
title('Phase spectrum')

%% power spectrum
yssq_sum_avg = sum(ys(1:nfft).^2)/nfft;
specsq_sum_avg = sum(abs(X).^2);

reference

OriginLab, How to Remove DC Offset before Performing FFT URL: http://blog.originlab.com/how-to-remove-dc-offset-before-performing-fft

How to remove DC component in FFT? URL: https://www.mathworks.com/matlabcentral/answers/712808-how-to-remove-dc-component-in-fft#answer_594373

Analyzing a signal that contains frequency content at Fs/2 doesn't seem to work unless there is a phase shift URL: https://dsp.stackexchange.com/a/59807/59253

Nyquist–Shannon sampling theorem, Critical frequency URL: https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem#Critical_frequency

Why remove energy at Nyquist before ifft? URL: https://dsp.stackexchange.com/a/22851/59253

1

emacs --no-site-file --load path/to/verilog-mode.el --batch filename.v -f verilog-auto-save-compile

CAUTION: filename.v is overwrite by command

verilog-mode.el

1
2
3
4

/*AUTOINPUT*/
/*AUTOWIRE*/
/*AUTOINST*/
/*AUTO_TEMPLATE*/

-f verilog-batch-auto

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:

reference

Emacs Online Documentation https://doc.endlessparentheses.com/

Emacs verilog-mode 的使用 URL: https://www.wenhui.space/docs/02-emacs/verilog_mode_useguide/

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 (
	input wire Trigger,
	input wire Pass,
	output reg A,
	output reg 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;


  lanhq1 C_reg ( .E(Trigger), .D(Pass), .Q(C) );
  an02d0 U3 ( .A1(Pass), .A2(Trigger), .Z(A) );
endmodule

add default value

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 (
	input wire Trigger,
	input wire Pass,
	output reg A,
	output reg 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;


  nd12d0 U5 ( .A1(Pass), .A2(Trigger), .ZN(C) );
  an02d0 U6 ( .A1(Pass), .A2(Trigger), .Z(A) );
endmodule

if evaluation

signed number cast to unsigned automatically before evaluating

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

// tb.v
module tb; 
  reg signed [1:0] datasg;
  reg [1:0] dataug;

  initial begin
    datasg = 2'b11;
    dataug = 2'b11;

    $display("datasg(%%d): %d", datasg);
    $display("dataug(%%d): %d", dataug);

    if (datasg)
      $display("datasg is OK");
    if (dataug)
      $display("dataug is OK");

    $finish();
  end 
endmodule

1
2
3
4
5
6
7
8
9

$ vlog tb.v
$ vsim -c -do "run;exit" work.tb
# Loading work.tb(fast)
# run
# datasg(%d): -1
# dataug(%d): 3
# datasg is OK
# dataug is OK
# ** Note: $finish    : tb.v(16)

reference

UC Berkeley CS150 Lec #20: Finite State Machines Slides

Lee WF. Learning from VLSI Design Experience [electronic Resource] / by Weng Fook Lee. 1st ed. 2019. Springer International Publishing; 2019. doi:10.1007/978-3-030-03238-8

• 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

Types of Hazards (on an output)

static 1-hazard, static 0-hazard, dynamic hazard

Hazard's Concern

• Hazards do not hurt synchronous circuits
• Hazards Kill Asynchronous Circuits
• Glitches Increase Power Consumption

referece

CPE166/EEE 270 Advanced Logic Design-Digital Design: Time Behavior of Combinational Networks: https://www.csus.edu/indiv/p/pangj/166/f/sram/Handout_Hazard.pdf

John Knight, ELEC3500 Glitches and Hazards in Digital Circuits http://www.doe.carleton.ca/~shams/ELEC3500/hazards.pdf

• Arithmetic operations have a potential to run into a condition known as overflow.
• Overflow occurs with respect to the size of the data type that must accommodate the result.
• Overflow indicates that the result was too large or too small to fit in the original data type.

Overflow when adding unsigned number

When two unsigned numbers are added, overflow occurs if

• there is a carry out of the leftmost bit.

Overflow when adding signed numbers

When two signed 2's complement numbers are added, overflow is detected if:

1. both operands are positive and the result is negative, or
2. both operands are negative and the result is positive.

Notice that when operands have opposite signs, their sum will never overflow. Therefore, overflow can only occur when the operands have the same sign.

carryout information ISN'T needed to detect overflow/underflow for signed number addition

EXTBIT:MSB

extended 1bit and msb bit can be used to detect overflow or underflow

1
2
3
4
5
6
7
8
9
10

reg signed  [1:0]      acc_inc;
reg signed  [10-1:0]   acc;
wire signed [10  :0]   acc_w;  // extend 1b for saturation
wire signed [10-1:0]   acc_stat;

assign acc_w = acc + acc_inc;   // signed arithmetic

assign acc_stat = (acc_w[10-1 +: 2] == 2'b01) ? {1'b0, {(10-1){1'b1}}} : // up saturation
    (acc_w[10-1 +: 2] == 2'b10) ? {1'b1, {(10-1){1'b0}}}  :   			 // down saturation
    acc_w[10-1:0];

2'b01 : overflow, up saturation

2'b10: underflow, down saturation

2's complement negative number

1. Flip all bits
2. Add 1.

N-bit signed number \[ A = -M_{N-1}2^{N-1}+\sum_{k=0}^{N-2}M_k2^k \] Flip all bits \[\begin{align} A_{flip} &= -(1-M_{N-1})2^{N-1} +\sum_{k=0}^{N-2}(1-M_k)2^k \\ &= M_{N-1}2^{N-1}-\sum_{k=0}^{N-2}M_k2^k -2^{N-1}+\sum_{k=0}^{N-2}2^k \\ &= M_{N-1}2^{N-1}-\sum_{k=0}^{N-2}M_k2^k -1 \end{align}\]

Add 1 \[\begin{align} A_- &= A_{flip}+1 \\ &= M_{N-1}2^{N-1}-\sum_{k=0}^{N-2}M_k2^k \\ &= -A \end{align}\]

reference

Overflow Detection: http://www.c-jump.com/CIS77/CPU/Overflow/lecture.html