snr

Signal-to-noise ratio

Syntax

r = snr(xi,y)

r = snr(x)

r = snr(x,fs,n)

r = snr(pxx,f,"psd")

r = snr(pxx,f,n,"psd")

r = snr(sxx,f,rbw,"power")

r = snr(sxx,f,rbw,n,"power")

r = snr(___,"aliased")

[r,noisepow] = snr(___)

snr(___)

Description

r = snr(xi,y) returns the signal-to-noise ratio (SNR) in decibels of a signal xi by computing the ratio of its summed squared magnitude to that of the noise y:

r = mag2db(rssq(xi(:))/rssq(y(:))).

y must have the same dimensions as xi. Use this form when the input signal is not necessarily sinusoidal and you have an estimate of the noise.

r = snr(x) returns the SNR in decibels relative to the carrier (dBc) of a real-valued sinusoidal input signal 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(x,fs,n) returns the SNR in dBc of a real sinusoidal input signal x sampled at a rate fs. The computation excludes the power contained in the lowest n harmonics, including the fundamental.

r = snr(pxx,f,"psd") specifies the input pxx as a one-sided power spectral density (PSD) estimate. The argument f is a vector of the frequencies at which the estimates of pxx occur. The computation of noise excludes the power of the first six harmonics, including the fundamental.

r = snr(pxx,f,n,"psd") specifies the number of harmonics n to exclude when computing the SNR. The default value of n is 6 and includes the fundamental.

r = snr(sxx,f,rbw,"power") specifies the input sxx as a one-sided power spectrum of a real signal. The input rbw is the resolution bandwidth over which each power estimate is integrated.

r = snr(sxx,f,rbw,n,"power") specifies the number of harmonics n to exclude when computing the SNR. The default value of n is 6 and includes the fundamental.

r = snr(___,"aliased") removes harmonics of the fundamental that are aliased into the Nyquist range. Use this option when the sinusoidal input signal is undersampled. If you do not specify this option, or if you set it to "omitaliases", then the function treats as noise any harmonics of the fundamental frequency that lie beyond the Nyquist range.

[r,noisepow] = snr(___) also returns the total noise power of the nonharmonic components of the signal.

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(xi,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 2/5 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.

Input Arguments

xi — Input signal

vector | matrix | N-D array

Input signal, specified as a vector, matrix, or N-D array.

Data Types: double | single
Complex Number Support: Yes

y — Noise estimate

vector | matrix | N-D array

Estimate of the noise in the input signal, specified as a vector, matrix, or N-D array with the same dimensions as xi.

Data Types: double | single
Complex Number Support: Yes

x — Real-valued sinusoidal input signal

vector

Real-valued sinusoidal input signal, specified as a row or column vector.

Data Types: double | single

fs — Sample rate

1 (default) | positive real scalar

Sample rate, specified as a positive scalar. The sample rate is the number of samples per unit time. If the unit of time is seconds, then the sample rate has units of hertz.

Data Types: double | single

n — Number of harmonics

6 (default) | positive integer scalar

Number of harmonics to exclude from the SNR computation, specified as a positive integer scalar. The default value of n is 6.

pxx — One-sided PSD estimate

vector

One-sided power spectral density estimate, specified as a real-valued, nonnegative column vector.

The power spectral density must be expressed in linear units, not decibels. Use db2pow to convert decibel values to power values.

Example: [pxx,f] = periodogram(cos(pi./[4;2]*(0:159))'+randn(160,2)) specifies the periodogram PSD estimate of a noisy two-channel sinusoid sampled at 2π Hz and the frequencies at which it is computed.

Data Types: double | single

f — Cyclical frequencies

real-valued row or column vector

Cyclical frequencies of the one-sided PSD estimate pxx, specified as a row or column vector. The first element of f must be 0.

Data Types: double | single

sxx — Power spectrum

nonnegative real-valued row or column vector

Power spectrum, specified as a real-valued nonnegative row or column vector.

The power spectrum must be expressed in linear units, not decibels. Use db2pow to convert decibel values to power values.

Example: [sxx,w] = periodogram(cos(pi./[4;2]*(0:159))'+randn(160,2),"power") specifies the periodogram power spectrum estimate of a two-channel sinusoid embedded in white Gaussian noise and the normalized frequencies at which it is computed.

Data Types: double | single

rbw — Resolution bandwidth

positive scalar

Resolution bandwidth, specified as a positive scalar. The resolution bandwidth is the product of the frequency resolution of the discrete Fourier transform and the equivalent noise bandwidth of the window.

Data Types: double | single

Output Arguments

r — Signal-to-noise ratio

real-valued scalar

Signal-to-noise ratio, returned as a real-valued scalar expressed in decibels relative to the carrier (dBc). The SNR is returned in decibels (dB) if the input signal is not sinusoidal.

Data Types: double | single

noisepow — Total noise power

real-valued scalar

Total noise power of the nonharmonic components of the input signal, returned as a real-valued scalar.

Data Types: double | single

More About

Distortion Measurement Functions

The functions thd, sfdr, sinad, and snr measure the response of a weakly nonlinear system stimulated by a sinusoid.

When given time-domain input, snr performs a periodogram using a Kaiser window with large sidelobe attenuation. To find the fundamental frequency, the algorithm searches the periodogram for the largest nonzero spectral component. It then computes the central moment of all adjacent bins that decrease monotonically away from the maximum. To be detectable, the fundamental should be at least in the second frequency bin. Higher harmonics are at integer multiples of the fundamental frequency. If a harmonic lies within the monotonically decreasing region in the neighborhood of another, its power is considered to belong to the larger harmonic. This larger harmonic may or may not be the fundamental.

The function estimates a noise level using the median power in the regions containing only noise. The DC component is excluded from the calculation. The noise at each point is the estimated level or the ordinate of the point, whichever is smaller. The noise is then subtracted from the values of the signal and the harmonics.

snr fails if the fundamental is not the highest spectral component in the signal.

Ensure that the frequency components are far enough apart to accommodate for the sidelobe width of the Kaiser window. If this is not feasible, you can use the "power" flag and compute a periodogram with a different window.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• If x and y are complex at compile time, then their size must be constant at compile time.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2013b

R2023a: Use gpuArray objects

The snr function supports gpuArray objects. You must have Parallel Computing Toolbox™ to use this functionality.