Brian Park, David Luong, Mark Piper, Aron Dobos
Lab 3, E71 Digital
Signal Processing,
Quantization Noise Effect
1. Quantization of Sinusoidal Signal (rounding to nearest integer bit)
The experimental and theoretical SQNR’s
were very similar. As the number of bits
increased, so did the SQNR as we would expect.
The graphs and the results in the table are generated using the Matlab script. Note: X denotes the original signal and xd is the quantized signal for all
graphs in this lab.
|
Bits |
Experimental SQNR
(dB) |
Theoretical SQNR
(dB) |
Mean Quantization
Noise |
Variance in
Quantization Noise |
|
4 |
26.201 |
25.84 |
2.89E-02 |
1.20E-03 |
|
8 |
49.7767 |
49.92 |
1.98E-03 |
5.17E-06 |
|
12 |
74.8614 |
74.00 |
1.07E-04 |
1.63E-08 |
|
16 |
98.0608 |
98.08 |
7.59E-06 |
7.85E-11 |
t=0:0.1:4*pi;
fprintf('Exp\t\tTh\t\tMean\t\tVar\n');
for n=4:
x=sin(t);
xd=round(x*2^(n-1))/2^(n-1);
error=xd-x;
error_mean=mean(abs(error));
error_variance=var(error);
figure; plot(t,x,t,xd,t,error)
legend('x','xd','error')
SQNR=10*log10(sum(x.^2)/sum(error.^2));
SQNR_eqn=1.76+6.02*n;
title(strcat(num2str(n),' bits. SQNR=',num2str(SQNR),' dB.')) fprintf('%0.4f\t%0.4f\t%0.3e\t%0.3e\n',SQNR,SQNR_eqn,error_mean,error_variance);
end
2. Quantization of Sinusoidal Signal (truncating to integer bit)
The experimental SQNR was always lower than the theoretical value. This was expected because instead of rounding the data as in part 1, we truncated, which leads to less accurate quantization of the signal and a lower SQNR.
|
Bits |
Experimental SQNR
(dB) |
Theoretical SQNR
(dB) |
Mean Quantization
Noise |
Variance in
Quantization Noise |
|
4 |
19.3759 |
25.84 |
6.54E-02 |
1.49E-03 |
|
8 |
44.0569 |
49.92 |
3.82E-03 |
5.06E-06 |
|
12 |
68.1629 |
74 |
2.29E-04 |
2.38E-08 |
|
16 |
91.9115 |
98.08 |
1.56E-05 |
7.97E-11 |
t=0:0.1:4*pi;
fprintf('Exp\t\tTh\t\tMean\t\tVar\n');
for n=4:
x=sin(t);
xd=floor(x*2^(n-1))/2^(n-1);
error=xd-x;
error_mean=mean(abs(error));
error_variance=var(error);
figure; plot(t,x,t,xd,t,error)
legend('x','xd','error')
SQNR=10*log10(sum(x.^2)/sum(error.^2));
SQNR_eqn=1.76+6.02*n;
title(strcat(num2str(n),' bits. SQNR=',num2str(SQNR),' dB.')) fprintf('%0.4f\t%0.4f\t%0.3e\t%0.3e\n',SQNR,SQNR_eqn,error_mean,error_variance);
end
3. Quantization of Gaussian Noise Signal
As expected, the SQNR tends to increase with the number of bits used to quantize the signal.
|
Bits |
Experimental SQNR
(dB) |
Mean Quantization
Noise |
Variance in
Quantization Noise |
|
4 |
29.1933 |
3.22E-02 |
1.34E-03 |
|
8 |
52.0926 |
1.86E-03 |
4.82E-06 |
|
12 |
77.1105 |
1.23E-04 |
1.99E-08 |
|
16 |
101.1026 |
7.33E-06 |
7.52E-11 |
t=0:0.1:4*pi;
fprintf('Exp\t\tTh\t\tMean\t\tVar\n');
for n=4:
x=randn(1,length(t));
xd=round(x*2^(n-1))/2^(n-1);
error=xd-x;
error_mean=mean(abs(error));
error_variance=var(error);
figure; plot(t,x,t,xd,t,error)
legend('x','xd','error')
SQNR=10*log10(sum(x.^2)/sum(error.^2));
SQNR_eqn=1.76+6.02*n;
title(strcat(num2str(n),' bits. SQNR=',num2str(SQNR),' dB.')) fprintf('%0.4f\t%0.4f\t%0.3e\t%0.3e\n',SQNR,SQNR_eqn,error_mean,error_variance);
end
4. Quantization of Sinusoidal Signal with Gaussian Random Noise
As we expected, the SNR is approximately 20dB because that is the specified effect of the Gaussian noise we added to the signal. The quantization noise decreases with increasing number of bits used and leads to a higher SQNR.
|
Bits |
Experimental SNR
(dB) |
Experimental SQNR
(dB) |
Theoretical SQNR
(dB) |
Mean Quantization
Noise |
|
4 |
19.0702 |
29.1933 |
25.84 |
3.22E-02 |
|
8 |
19.7978 |
52.0926 |
49.92 |
1.86E-03 |
|
12 |
19.646 |
77.1105 |
74 |
1.23E-04 |
|
16 |
19.4478 |
101.1026 |
98.08 |
7.33E-06 |
t=0:0.1:4*pi;
fprintf('SNR\t\tExp\t\tTh\t\tMean\n');
for n=4:
x_original=sin(t);
noise=randn(1,length(t))./sqrt(200);
x=x_original+noise;
xd_original=round(x_original*2^(n-1))/2^(n-1);
xd=round(x*2^(n-1))/2^(n-1);
error_original=xd_original-x_original;
error=xd-x;
error_mean=mean(abs(error));
figure; plot(t,x,t,xd,t,error)
legend('x','xd','error')
SNR=10*log10(sum(x.^2)/(sum(error.^2)+sum(noise.^2)));
SQNR=10*log10(sum(x.^2)/(sum(error.^2)));
SQNR_eqn=1.76+6.02*n;
title(strcat(num2str(n),' bits. SNR=',num2str(SNR),' dB.')) fprintf('%0.4f\t%0.4f\t%0.4f\t%0.3e\n',SNR,SQNR,SQNR_eqn,error_mean);
end