snr
Signal-to-noise ratio
Syntax
Description
r = snr(x,y)x,
by computing the ratio of its summed squared magnitude to that of
the noise, y. y must have
the same dimensions as x. Use this form when
the input signal is not necessarily sinusoidal and you have an estimate
of the noise.
r = snr(x)x. The SNR is determined
using a modified periodogram of the same length as the input. The
modified periodogram uses a Kaiser window with β =
38. The result excludes the power of the first six harmonics, including
the fundamental.
r = snr(___,'aliased')'omitaliases',
then the function treats as noise any harmonics of the fundamental
frequency that lie beyond the Nyquist range.
snr(___) with no output arguments
plots the spectrum of the signal in the current figure window and
labels its main features. It uses different colors to draw the fundamental
component, the DC value and the harmonics, and the noise. The SNR
appears above the plot. This functionality works for all syntaxes
listed above except snr(x,y).
Examples
Signal-to-Noise Ratio for Rectangular Pulse with Gaussian Noise
Generate a 20-millisecond rectangular pulse sampled for 2 seconds at 10 kHz.
Tpulse = 20e-3; Fs = 10e3; t = -1:1/Fs:1; x = rectpuls(t,Tpulse);
Embed the pulse in white Gaussian noise such that the signal-to-noise ratio (SNR) is 53 dB. Reset the random number generator for reproducible results.
rng default
SNR = 53;
y = randn(size(x))*std(x)/db2mag(SNR);
s = x + y;Use the snr function to compute the SNR of the noisy signal.
pulseSNR = snr(x,y)
pulseSNR = 53.1255
Compare SNR with THD and SINAD
Compute and compare the signal-to-noise ratio (SNR), the total harmonic distortion (THD), and the signal to noise and distortion ratio (SINAD) of a signal.
Create a sinusoidal signal sampled at 48 kHz. The signal has a fundamental of frequency 1 kHz and unit amplitude. It additionally contains a 2 kHz harmonic with half the amplitude and additive noise with variance 0.1².
fs = 48e3;
t = 0:1/fs:1-1/fs;
A = 1.0;
powfund = A^2/2;
a = 0.4;
powharm = a^2/2;
s = 0.1;
varnoise = s^2;
x = A*cos(2*pi*1000*t) + ...
a*sin(2*pi*2000*t) + s*randn(size(t));Verify that SNR, THD, and SINAD agree with their definitions.
SNR = snr(x); defSNR = 10*log10(powfund/varnoise); SN = [SNR defSNR]
SN = 1×2
17.0178 16.9897
THD = thd(x); defTHD = 10*log10(powharm/powfund); TH = [THD defTHD]
TH = 1×2
-7.9546 -7.9588
SINAD = sinad(x); defSINAD = 10*log10(powfund/(powharm+varnoise)); SI = [SINAD defSINAD]
SI = 1×2
7.4571 7.4473
Signal-to-Noise Ratio of Sinusoid
Compute the SNR of a 2.5 kHz sinusoid sampled at 48 kHz. Add white noise with variance 0.001².
Fi = 2500; Fs = 48e3; N = 1024; x = sin(2*pi*Fi/Fs*(1:N)) + 0.001*randn(1,N); SNR = snr(x,Fs)
SNR = 57.7103
Plot the spectrum and annotate the SNR.
snr(x,Fs);
SNR of Sinusoid Using PSD
Obtain the periodogram power spectral density (PSD) estimate of a 2.5 kHz sinusoid sampled at 48 kHz. Add white noise with standard deviation 0.00001. Use this value as input to determine the SNR. Set the random number generator to the default settings for reproducible results.
rng default Fi = 2500; Fs = 48e3; N = 1024; x = sin(2*pi*Fi/Fs*(1:N)) + 0.00001*randn(1,N); w = kaiser(numel(x),38); [Pxx, F] = periodogram(x,w,numel(x),Fs); SNR = snr(Pxx,F,'psd')
SNR = 97.7446
SNR of Sinusoid Using Power Spectrum
Using the power spectrum, compute the SNR of a 2.5 kHz sinusoid sampled at 48 kHz and embedded in white noise with a standard deviation of 0.00001. Reset the random number generator for reproducible results.
rng default Fi = 2500; Fs = 48e3; N = 1024; x = sin(2*pi*Fi/Fs*(1:N)) + 0.00001*randn(1,N); w = kaiser(numel(x),38); [Sxx, F] = periodogram(x,w,numel(x),Fs,'power'); rbw = enbw(w,Fs); SNR = snr(Sxx,F,rbw,'power')
SNR = 97.7446
Plot the spectrum of the signal and annotate the SNR.
snr(Sxx,F,rbw,'power');
SNR with and Without Aliased Harmonics
Generate a signal that resembles the output of a weakly nonlinear amplifier with a 2.1 kHz tone as input. The signal is sampled for 1 second at 10 kHz. Compute and plot the power spectrum of the signal. Use a Kaiser window with β = 38 for the computation.
Fs = 10000;
f = 2100;
t = 0:1/Fs:1;
x = tanh(sin(2*pi*f*t)+0.1) + 0.001*randn(1,length(t));
periodogram(x,kaiser(length(x),38),[],Fs,'power')
Harmonics stick out from the noise at frequencies of 4.2 kHz, 6.3 kHz, 8.4 kHz, 10.5 kHz, 12.6 kHz, and 14.7 kHz. All frequencies except for the first one are greater than the Nyquist frequency. The harmonics are aliased respectively into 3.7 kHz, 1.6 kHz, 0.5 kHz, 2.6 kHz, and 4.7 kHz.
Compute the signal-to-noise ratio of the signal. By default, snr treats the aliased harmonics as part of the noise.
snr(x,Fs,7);
Repeat the computation, but now treat the aliased harmonics as part of the signal.
snr(x,Fs,7,'aliased');
Noise Power
Create a sinusoidal signal sampled at 48 kHz. The signal has a fundamental of frequency 1 kHz and unit amplitude. It additionally contains a 2 kHz harmonic with half the amplitude and additive noise with variance 0.1².
fs = 48e3;
t = 0:1/fs:1-1/fs;
A = 1.0;
powfund = A^2/2;
a = 0.4;
powharm = a^2/2;
s = 0.1;
varnoise = s^2;
x = A*cos(2*pi*1000*t) + ...
a*sin(2*pi*2000*t) + s*randn(size(t));Compute the noise power in the signal. Verify that it agrees with the definition.
[SNR,npow] = snr(x,fs); compare = [10*log10(powfund)-npow SNR]
compare = 1×2
17.0281 17.0178
SNR of Amplified Signal
Generate a sinusoid of frequency 2.5 kHz sampled at 50 kHz. Reset the random number generator. Add Gaussian white noise with standard deviation 0.00005 to the signal. Pass the result through a weakly nonlinear amplifier. Plot the SNR.
rng default
fs = 5e4;
f0 = 2.5e3;
N = 1024;
t = (0:N-1)/fs;
ct = cos(2*pi*f0*t);
cd = ct + 0.00005*randn(size(ct));
amp = [1e-5 5e-6 -1e-3 6e-5 1 25e-3];
sgn = polyval(amp,cd);
snr(sgn,fs);
The DC component and all harmonics, including the fundamental, are excluded from the noise measurement. The fundamental and harmonics are labeled.