FFT of quantized signal

Question

Maruthi prasanna C 2020-2-2

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评论： Addy 2022-8-17

I am having difficulty in plotting FFT of quantized signal of an ideal qunatizer in MATLAB . When I plot the FFT in dB Scale , I am getting infinity as the value & figure is blank . Please help me, I have posted the Code below

fs  = 1e6;
fx  = 50e3;
Afs = 1;
N   = 2^11;
% time vector
t = linspace(0, (N-1)/fs, N);
% input signal
y = Afs * cos(2*pi*fx*t);
% spectrum
s = 20 * log10(abs(fft(y)/N/Afs*2));
% drop redundant half
s = s(1:N/2);
% frequency vector (normalized to fs)
f = (0:length(s)-1);
figure(1)
plot(y);
title('input signal ')
ylabel('y');
xlabel('frequency bins');
B = 2; % no of bits 
delta = 2/2^B; 
fs = 1e6;
% Number of cycles in test
cycles = 67; 
%Make N/cycles non-integer! accomplished by choosing cyclesÆprime #
%N=power of 2 speeds up analysis
% N = 2^10;
%signal frequency
fx = fs*cycles/N
y = Afs * cos(2*pi*fx*t);
s = 20 * log10(abs(fft(y)/N/Afs*2));
f = (0:length(s)-1);
figure(2)
plot(f,s);
title('spectrum of input signal ');
ylabel('y')
xlabel('frequency bins')
y1=round(y/delta)*delta;
figure(3)
plot(y,'b');
hold on
stem(y1,'r')
hold on
plot(y-y1,'g')
title(sprintf('Signal, Quantized signal and Error for %g bits, %g quantization levels',B,2^B));
xlabel('frequency');
hold off
y2=20*log10((abs(fft(y1)/N*2)));
figure(10)
f1 = (0:length(y2)-1);
plot(f1 ,y2)

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Sanjaya Senavirathne 2021-8-11

problem in this case is some values contains 0, so the FFT value of 0 is which is un imagineble that's why figure is blank. so i would suggest please add really very low noise like e-30 , then you will be able to see an output.

Chunru 2021-8-11

In theory, quantization can have any integer number of levels (not necessarily 2^B, though 2^B is prefered). The quantization can be non-uniform as well.

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Answer 1

Chunru 2021-8-11

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Quantizing the ideal sinusoidal signal will produce a perodical signal, which may have some weird spectrum with 0 at some frequencies causing the display problem. To study the quatization effect, you can use some other signals, for example chirp, pink noise. The following code add in an offset so that the spectrum is no longer "weird".

fs = 1e6;

fx = 50e3;

Afs = 1;

N = 2^11;

% time vector

t = linspace(0, (N-1)/fs, N);

% input signal

y = Afs * cos(2*pi*fx*t);

% spectrum

s = 20 * log10(abs(fft(y)/N/Afs*2));

% drop redundant half

s = s(1:N/2);

% frequency vector (normalized to fs)

f = (0:length(s)-1);

figure(1)

plot(y);

title('input signal ')

ylabel('y');

xlabel('frequency bins');

B = 2; % no of bits

delta = 2/2^B;

fs = 1e6;

% Number of cycles in test

cycles = 167;

%Make N/cycles non-integer! accomplished by choosing cyclesÆprime #

%N=power of 2 speeds up analysis

% N = 2^10;

%signal frequency

fx = fs*cycles/N

fx = 8.1543e+04

y = Afs * cos(2*pi*fx*t) + 0.123; % add a small offset

s = 20 * log10(abs(fft(y)/N/Afs*2));

f = (0:length(s)-1);

figure(2)

plot(f,s);

title('spectrum of input signal ');

ylabel('y')

xlabel('frequency bins')

y1=round(y/delta)*delta; y1'

ans = 2048×1

1.0000 1.0000 0.5000 0 -0.5000 -0.5000 -1.0000 -1.0000 -0.5000 0

figure(3)

plot(y,'b');

hold on

stem(y1,'r')

hold on

plot(y-y1,'g')

title(sprintf('Signal, Quantized signal and Error for %g bits, %g quantization levels',B,2^B));

xlabel('frequency');

hold off

y2=20*log10((abs(fft(y1)/N*2)))'

y2 = 2048×1

-13.8447 -58.6345 -62.4346 -74.3996 -103.6032 -59.0645 -60.1435 -59.3283 -45.3150 -38.9921

%y2=((abs(fft(y1)/N*2)))'

figure(10)

f1 = (0:length(y2)-1);

plot(f1 ,y2)

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Answer 2

Yazan 2021-8-11

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There are some issues with your code. See my modifications and comments below. You cannot calculate SNR, because you did not add noise to any signal (unless I'm missing something here).

clc, clear, close all

fs = 1e6;

fx = 50e3;

Afs = 1;

N = 2^11;

% time vector

t = linspace(0, (N-1)/fs, N);

% input signal

y = Afs * cos(2*pi*fx*t);

figure('Units', 'normalized', 'Position', [0.05 0.15 0.85 0.7])

subplot(3,2,1)

plot(t, y), title('Signal y'); axis tight

xlabel('Time - sec');

ylabel('Amplitude');

% spectrum

Nfft = N;

s = 1/Nfft * abs(fft(y, Nfft));

f = linspace(0, fs/2, Nfft/2+1);

subplot(3,2,2)

s = s(1:Nfft/2+1).^2;

sy = [s(1), 2*s(2:end-1), s(end)];

plot(f, pow2db(sy)), grid minor

title('One-sided power spectrum of y'), xlabel('Frequency - Hz');

ylabel('dB')

% or you can use the periodogram function with rect window

% you'll get exactly the same result

% [s2, f2] = periodogram(y, rectwin(length(y)), Nfft, fs, 'onesided', 'power');

% hold on, plot(f2, pow2db(s2))

% signal 2

B = 2;

delta = 2/2^B;

fs = 1e6;

cycles = 67;

fx = fs*cycles/N;

yy = Afs * cos(2*pi*fx*t);

subplot(3,2,3)

plot(t, yy), title('Signal yy'); axis tight

xlabel('Time - sec');

ylabel('Amplitude');

% spectrum

Nfft = N;

s = 1/Nfft * abs(fft(yy, Nfft));

s = s(1:Nfft/2+1).^2;

syy = [s(1), 2*s(2:end-1), s(end)];

subplot(3,2,4)

plot(f, pow2db(syy)), grid minor

title('One-sided power spectrum of yy'), xlabel('Frequency - Hz');

ylabel('dB')

% quantized signal

y1 = round(y/delta)*delta;

subplot(3,2,5)

n = 100;

plot(t(1:n), y1(1:n)), title('Signal y1'); axis tight

xlabel('Time - sec');

ylabel('Amplitude'); grid minor

hold on,

plot(t(1:n), y(1:n), '--')

err = y-y1;

plot(t(1:n), err(1:n))

legend({'Quantized', 'Original', 'Error'})

s = 1/Nfft * abs(fft(y1, Nfft));

s = s(1:Nfft/2+1).^2;

sy1 = [s(1), 2*s(2:end-1), s(end)];

subplot(3,2,6)

plot(f, pow2db(sy1)), grid minor

title('One-sided power spectrum of y1'), xlabel('Frequency - Hz');

ylabel('dB')

% estimate power

pRMSy = mag2db(rms(y));

pRMSyy = mag2db(rms(yy));

pRMSy1 = mag2db(rms(y1));

fprintf('Power of signal y = %g dBW\n', pRMSy)

Power of signal y = -3.01267 dBW

fprintf('Power of signal yy = %g dBW\n', pRMSyy)

Power of signal yy = -3.0103 dBW

fprintf('Power of signal y1 = %g dBW\n', pRMSy1)

Power of signal y1 = -2.22132 dBW

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Addy 2022-8-17

Hey in your code, 2 bit quantization, there are 5 levels.

you can see when you do "unique(y1)"

But for 2 bit quantization, there is only 4 levels. 00, 01, 10, 11.

So, dividing your signal into 4 parts is necessary (At least that is what I am understanding. Please correct me if I am wrong. I am still learning.

Otherwise, your code looks great.

I was working on that 4 level quantization.

clear;clc

A = 4;

f = 10e3;

% fs = 25e6; ts = 1/fs;

ts = 0.2/10/f(end);

t_end = 1/f;

t = 0:ts:t_end;

x = A*sin(2*pi*f*t)';

% x = awgn(x,100);

t = t*1e6;

%%

q_bits = 2;

q_num_levels = 2^q_bits;

q_levels = linspace(min(x),max(x),q_num_levels);

bins = interp1(1:numel(q_levels),q_levels,0.5 + (1:numel(q_levels)-1));

y = discretize(x,[-Inf bins Inf]);

y = normalize(y,'range',[min(q_levels),max(q_levels)]);

y = y-mean(y);

u = unique(y);

%%

bit_levels = 0:q_num_levels-1;

bit_levels = dec2bin(bit_levels,q_bits);

q = diff(q_levels); q = q(1);

q_error = (max(x)-min(x))/(2*q_num_levels);

q_noise_pwr = q/sqrt(12);

%%

figure(1);clf

hold on;box on

set(gcf,'color','k')

set(gca,'color','k')

set(gca, 'YGrid', 'on', 'XGrid', 'off')

set(gca,'GridColor','w')

set(gca,'XColor','w')

set(gca,'YColor','w')

set(gca,'GridAlpha',0.5)

plot(t,x,'c')

plot(t,y,'m');

plot(t,x-y,'-.g')

yticks(q_levels);

yticklabels(bit_levels);

yticklabels([]);

ylim([min(x)+min(x)/10,max(x)+max(x)/10])

xlim([min(t),max(t)])

tit = title(['Example with ',num2str(q_bits),' Bits[',num2str(q_num_levels),' levels] Linear Quantization']);

lgnd = legend( {'\color{cyan} Analog signal',...

'\color{magenta} Quantized signal',...

['\color{green} Quantized Error signal.',newline,...

' Max.Quant.Error = ',num2str(q_error)]});

set(lgnd, 'Color', 'k');

set(tit, 'Color', 'w');

ylabel('Quantization levels')

xlabel('Time - [µs]')

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FFT of quantized signal

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