Effective Noise BandWidth and Noise Floor

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PS and PSD

The spectral density format is appropriate for random or noise signals but inappropriate for discrete frequency components because the latter theoretically have zero bandwidth

Amplitude Correction

  1. A finite-duration window \(w[n]\)

    DTFT is \(W[e^{j\omega}]\) and the maximum magnitude is is at DC frequency, which \(\sum_n w_n\)

  2. Sinusoidal signal \(x[n]\)

    DFS is \(X_k\), and DTFT shall be \(\frac{2\pi}{N}X_k(e^{j\omega})\)

  3. the windowed sequence \(v[n] = x[n]w[n]\)

    with multiplication property, DTFT of \(v[n]\) shall be \(\frac{X_k(e^{j\omega})}{N}\sum_n w_n\)

    As we know, DFT of \(v[n]\) is samples of its DTFT, that is \[ \frac{X_k(e^{j\omega})}{N}\sum_n w_n = X_v[k] \] Therefore, \[ \frac{X_k(e^{j\omega})}{N} = \frac{X_v[k]}{\sum_n w_n} \]

Effective Noise BandWidth (ENBW)

General derivation

The relationship between a power spectrum (\(PS, V^2\)) and a power spectral density (\(PSD, V^2/Hz\)) is given by the effective noise bandwidth (ENBW), which can easily be determined at the time when the DFT is computed.

ENBW should always be recorded when a spectrum or spectral density is computed, such that the result can be converted to the other form at a later stage, when the information about the frequency resolution \(f_{res}\) and the window that was used is normally not easily available any more.

The normalized equivalent noise bandwidth (NENBW) of the window is given by

\[ \text{NENBW} = \frac{NS_2}{S_1^2} \] where \(S_1 = \sum _{k=0}^{N-1}w_k\) and \(S_2 = \sum _{k=0}^{N-1}w_k^2\)

The ENBW is given by

\[ \text{ENBW} = \text{NENBW}\cdot f_{res} = \text{NENBW}\cdot \frac{f_s}{N} = f_s\frac{S_2}{S_1^2} \]

Equivalent noise bandwidth (ENBW) compares a window to an ideal, rectangular time-window. It is the bandwidth of the rectangular window's frequency-domain shape that passes the same amount of white noise energy as the frequency-domain shape defined by the other window.

enbw

Therefore, the equivalent noise bandwidth \(B_{enbw}\) is given by

\[ B_{enbw} = \frac{\int_{-f}^{f} |W(f)|^2 df}{|W(f_0)|^2} \]

Translating to discrete domain, the equivalent noise bandwidth can be computed using DFT samples as

\[ B_{enbw} =\frac{\sum_{k=0}^{N-1}|W[k]|^2}{|W[k_0]|^2} \]

where \(k_0\) is the index at which the magnitude of FFT output is maximum and \(N\) is the window length, i.e. \(k_0=0\).

Applying Parseval's theorem and \(W[0]=\sum_{n}w[n]\), \(B_{enbw}\) can also be computed using time domain samples as

\[ B_{enbw} = N \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2} \]

scale \(w[n]\) don't change \(B_{enbw}\)

Noise power inside window: \(\int_{-f}^{f} |W(f)|^2 df \to N\cdot\sum_{n}|w[n]|^2\)

peak amplitude: \(|W(f_0)|^2 \to \left| \sum_{n} w[n] \right|^2\)

An alternative derivation

coherence-incoherence.drawio.svg

Assuming the windowed sequence \(v[n] = x[n]w[n]\)

  • \(W[k]\): Fourier Transform of finite sequence window

  • \(X_{sig}\): Fourier Transform of signal

  • \(X_{n}\): Fourier Transform of noise

  • \(X_{v,sig}\): Fourier Transform of windowed signal

  • \(X_{v,n}\): Fourier Transform of windowed noise

From Fig. 6,, we observe that the amplitude of the harmonic estimate at a given frequency is biased by the accumulated broad-band noise included in the bandwidth of the window.

image-20240517195407690

In this sense, the window behaves as a filter, gathering contributions for its estimate over its bandwidth

The Fourier Transform of windowed signal can be expressed as

\[\begin{align} X_{v,sig} &= W_{max}\cdot X_{sig} \\ &= W[0]\cdot X_{sig} \end{align}\]

For a typical window, \(W_{max}\) occurs at \(\omega = 0\)

And the Fourier Transform of windowed noise can be expressed as

\[ X_{v,n}^2 = \sum_k (W[k])^2 \cdot X_n^2 \]

divided by \((W[0])^2\) on both sides of the above equation

\[ \frac{X_{v,n}^2}{(W[0])^2} = \frac{\sum_k (W[k])^2}{(W[0])^2} \cdot X_n^2 \]

By Parseval's theorem

\[ \frac{X_{v,n}^2}{\left(\sum_n w[n]\right)^2} = \frac{N\sum_n w^2[n]}{\left(\sum_n w[n]\right)^2} \cdot X_n^2 \]

where \(X^2_n\) is what is deserved and

\[ X_n^2 = \frac{PS_{n}}{B_{enbw}} \]

where \(B_{enbw} = N \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2}\) and \(PS_{n}=\left| \frac{X_{v,n}}{\sum_n w[n]}\right|^2\)

code example

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lw = 128;
win = hann(lw);
lt = 2048;
windft = fftshift(fft(win,lt));

ad = abs(windft).^2;
mg = max(ad);

fs = 1000;

bw = enbw(win,fs);

bdef = sum((win).^2)/sum(win)^2*fs;
fprintf("bw: %.3f\n", bw);
fprintf("bdef: %.3f\n", bdef);

freq = -fs/2:fs/lt:fs/2-fs/lt;

plot(freq,ad, bw/2*[-1 -1 1 1],mg*[0 1 1 0],'--')
xlim(bw*[-1 1])

Adiff = trapz(freq,ad)-bw*mg;
fprintf("Adiff: %.3e\n", Adiff);

Verify that the area of the rectangle contains the same total power as the window. 

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Adiff = trapz(freq,ad)-bw*mg
output: 
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bw: 11.811
bdef: 11.811
Adiff: 7.276e-12

image-20230530230434715

Noise Floor

General Formula

signal tone power \[ P_{\text{sig}} = 2 \frac{X_{w,sig}^2}{S_1^2} \]

noise power \[ P_n = \frac{X_{w,n}^2}{S_2} \]

Then, displayed SNR is obtained \[\begin{align} \mathrm{SNR} &= 10\log10\left(\frac{X_{w,sig}^2}{X_{w,n}^2}\right) \\ &= 10\log_{10}\left(\frac{P_{\text{sig}}}{P_n}\right) + 10\log_{10}\left(\frac{S_1^2}{2S_2}\right) \\ &= \mathrm{SNR}'-10\log_{10}\left(\frac{2S_2}{S_1^2}\right) \\ \end{align}\]

DFT's output \(\mathrm{SNR}\)

Rect Window

DFT bin's output noise standard deviation (rms) value is proportional to \(\sqrt{N}\), and the DFT's output magnitude for the bin containing the signal tone is proportional to \(N\)

signal tone power \[ P_{\text{sig}} = 2 \frac{X_{w,sig}^2}{N^2} \]

+/- frequency spectrum and apply Parseval's theorem

noise power \[\begin{align} P_n &= \sum_{k=0}^{N-1}{\frac{(X_{w,n}(k))^2}{N^2}} \\ &= \frac{X_{w,n}^2}{N^2}\cdot N \\ &= \frac{X_{w,n}^2}{N} \end{align}\]

note white noise, that is \(X_n(i) = X_n(j)\) for any \(i \neq j\)

displayed SNR \[\begin{align} \mathrm{SNR} &= 10\log10\left(\frac{X_{w,sig}^2}{X_{w,n}^2}\right) \\ &= 10\log_{10}\left(\frac{P_{\text{sig}}N^2}{P_n2N}\right) \\ &= 10\log_{10}\left(\frac{P_{\text{sig}}}{P_n}\right) + 10\log_{10}\left(\frac{N}{2}\right) \\ &= \mathrm{SNR}' - 10\log_{10}(2/N) \end{align}\]

If we increase a DFT's size from \(N\) to \(2N\), the DFT's output SNR increased by 3dB. So we say that a DFT's processing gain increases by 3dB whenever \(N\) is doubled.

[http://individual.utoronto.ca/schreier/lectures/2015/1.pdf]

No. of nonzero FFT bins

The problem with sine-wave scaling is that the noise power is, on average, evenly distributed over all FFT bins, whereas the sine-wave power is concentrated in only a few bins. With sine-wave scaling, the power of individual sine-wave components can be read directly from the spectral plot, but in order to determine the noise power, the powers of all the noise bins must be added together.

snr_final.drawio

We can't use Fourier transform of random signal

[https://raytroop.github.io/2023/11/10/random/#lti-systems-on-wss-processes]

\[\begin{align} \mathrm{SNR} &= \frac{X_\text{sig}^2}{X_\text{n}^2N} \\ &= \frac{X_\text{sig}^2\cdot \sum_k W_k^2}{X_\text{n}^2N\cdot \sum_k W_k^2} \\ &= \frac{\sum_\text{nb} X_\text{w,sig}^2}{N X_\text{w,n}^2} \end{align}\]

The number of nonzero signal bins for the Hann window is \(\mathrm{nb} = 3\)

image-20240522213736493

where \(\mathrm{NBW} = \frac{\sum_{n}|w[n]|^2}{ \left| \sum_{n} w[n] \right|^2}\)

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% excerpt of A.4 An Example in Pavan, Schreier and Temes, "Understanding Delta-Sigma Data Converters, Second Edition" ISBN 978-1-119-25827-8
%
% Compute modulator output and actual NTF
%
OSR = 32;
ntf0 = synthesizeNTF(5,OSR,1);
N = 64*OSR;
fbin = 11;
u = 1/2*sin(2*pi*fbin/N*[0:N-1]);
[v tmp1 tmp2 y] = simulateDSM(u,ntf0);
k = mean(abs(y)/mean(y.^2))
ntf = ntf0 / (k + (1-k)*ntf0);
%
% Compute windowed FFT and NBW
%
w = hann(N); % or ones(1,N) or hann(N).^2
nb = 3; % 1 for Rect; 5 for Hann^2
w1 = norm(w,1);
w2 = norm(w,2);
NBW = (w2/w1)^2
V = fft(w.*v)/(w1/2);
%
% Compute SNR
%
signal_bins = fbin + [-(nb-1)/2:(nb-1)/2];
inband_bins = 0:N/(2*OSR);
noise_bins = setdiff(inband_bins,signal_bins);
snr = dbp(sum(abs(V(signal_bins+1)).^2)/sum(abs(V(noise_bins+1)).^2))

demo

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%https://aaronscher.com/Course_materials/Communication_Systems/documents/PSD_Autocorrelation_Noise.pdf

%Clear variables. clear command window, close all figures:
clc;
clear all;
close all;
%%%Setup and define variables
f0=10; %frequency of sinusoidal signal (Hz)
fs=100; %sampling frequency (Hz)
Ts=1/fs; %sampling period (seconds)
N0=3000; %number of samples
t=[0:Ts:Ts*(N0-1)]; %Sample times
noise_PSD=.5; %This is the desired noise power spectral density in W/Hz.
variance=noise_PSD*fs;% Variance = sigma^2
sigma=sqrt(variance);
noise=transpose(sigma*randn(N0,1));%create sampled white Gaussian noise.
xsignal=20*sin(2*pi*f0*t); %create sampled sinusoidal signal
x=xsignal+noise; %Add signal to noise
figure(1)
histogram(noise,30) %plot histogram
set(gca,'FontSize',14) %set font size of axis tick labels to 18
xlabel('Noise amplitude','fontsize',14)
ylabel('Frequency of occurance','fontsize',14)
title('Simulated histogram of white Gaussian noise','fontsize',14)
SNR_try1=snr(xsignal,noise); %calculate SNR using built in "snr" function.
SNR_try2=10*log10(sum(xsignal.^2)/sum(noise.^2)); %manually calculate SNR.
%If everything is correct, the two SNR calculations above should agree.
%Plot noise in time-domain
figure(2)
plot(t,x)
set(gca,'FontSize',14) %set font size of axis tick labels to 18
xlabel('Time (s)','fontsize',14)
ylabel('Amplitude','fontsize',14)
title('Noisey sinusoid','fontsize',14)
grid on
%Plot power spectral density (PSD) of noise using three different methods:
%
%Method 1. Calcululate PSD from amplitude spectrum
N=2^16; %Number of discrete points in the FFT
y=fft(x,N)/fs; %fft of noise
z=fftshift(y);%center noise spectrum
f_vec=[0:1:N-1]*fs/N-fs/2; %designate sample frequencies
amplitude_spectrum=abs(z); %compute two-sided amplitude spectrum
ESD1=amplitude_spectrum.^2; %ESD = |F(w)|^2;
PSD1=ESD1/((N0-1)*Ts);% PSD=ESD/T where T = total time of sample
figure(3)
plot(f_vec,10*log10(PSD1));
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 1','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
%calculate average power using PSD calclated from method 1:
Average_power_method_1=sum(PSD1)*fs/N; % Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Method 2 - Calculate PSD from autocorrelation
time_lag=((-length(x)+1):1:(length(x)-1))*Ts;
auto_cor=xcorr(x,x)/fs; %Use xcorr function to find PSD
y=1/fs*fft(auto_cor,N); %fft of auto correlation function
PSD2=abs(1/(N0-1)*fftshift(fft(auto_cor,N)));
figure(4)
plot(f_vec,10*log10(PSD2));%use convolution
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 2','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
%calculate average power using PSD calclated from method 1:
Average_power_method_2=sum(PSD2)*fs/N; %Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Method 3 - Calculate PSD using built in pwelch function
figure(5)
PSD3=periodogram(x,[],N,fs,'centered');
plot(10*log10(PSD3))
xlabel('Frequency [Hz]','fontsize',14)
ylabel('dB/Hz','fontsize',14)
title('Power spectral density - method 3','fontsize',14)
grid on
set(gcf,'color','w'); %set background color from grey (default) to white
axis tight
Average_power_method_3=sum(PSD3)*fs/N; %Pav=sum(PSD)*delta_f where delta_f=fs/N;
%
%Calculate mean and average PSD of noise:
PSD_noise=periodogram(noise,[],N,fs,'centered');
Average_noise_PSD=mean(PSD_noise);
Mean_noise=mean(noise);

image-20230531004822838

image-20230531005008418

The power spectral density plots for methods 2 and 3 exactly match that for method 1 (shown above).

reference

Schmid, H. (2012). How to use the FFT and Matlab's pwelch function for signal and noise simulations and measurements. URL:http://www.schmid-werren.ch/hanspeter/publications/2012fftnoise.pdf

Bonnie C.Baker. Reading and Using Fast Fourier Transforms (FFT) URL:http://ww1.microchip.com/downloads/en/appnotes/00681a.pdf

FFT analysis: SNR and Noise level vary with FFT Window size URL:https://www.virtins.com/forum/viewtopic.php?f=7&t=1382

Lyons, R. G. (2011). Understanding digital signal processing (3rd ed.). Prentice Hall

Stefan Scholl, "Exact Signal Measurements using FFT Analysis",Microelectronic Systems Design Research Group, TU Kaiserslautern, Germany. [pdf]

enbw Matlab URL:https://www.mathworks.com/help/signal/ref/enbw.html

Aaron Scher. PSD, Autocorrelation, and Noise in MATLAB [pdf]

Aaron Scher. FFT, total energy, and energy spectral density computations in MATLAB [pdf]

Amplitude Estimation and Zero Padding URL:https://www.mathworks.com/help/signal/ug/amplitude-estimation-and-zero-padding.html

Harris, F. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66, 51-83. [pdf]

Harris, F. J. . (1976). Windows, harmonic analysis and the discrete fourier transform. [pdf]

Wang Hongwei. Virtins. Evaluation of Various Window Functions using Multi-Instrument [pdf]

Properties of FFT Windows Used in Stable32 [pdf]

Solomon, Jr, O M. PSD computations using Welch's method. [Power Spectral Density (PSD)]. United States. https://doi.org/10.2172/5688766 [pdf]

Measure Power of Deterministic Periodic Signals [https://www.mathworks.com/help/signal/ug/measuring-the-power-of-deterministic-periodic-signals.html]

Mathuranathan. Equivalent noise bandwidth (ENBW) of window functions. [https://www.gaussianwaves.com/2020/09/equivalent-noise-bandwidth-enbw-of-window-functions/]

recordingblogs, Equivalent noise bandwidth [https://www.recordingblogs.com/wiki/equivalent-noise-bandwidth]

unpingco, Python for Signal Processing [https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Windowing_Part2.ipynb]

Pavan, Schreier and Temes, "Understanding Delta-Sigma Data Converters, Second Edition" ISBN 978-1-119-25827-8