Window Functions Archives

Key focus: Window function smooths the observed signal over the edges. Analysis of some important parameters to help select the window for an application.

Spectral leakage

As we know, the DFT operation can be viewed as processing a signal through a set of filter banks with bandwidth Δf centered on the bin (frequency) of interest (Figure 1). To accurately resolve the frequency component in each bin, we desire the resolution bandwidth Δf to be as small as possible. In effect, each filter filter bank is viewed as filter having a brick-wall response with narrow bandwidth. If the signal under study contains several spectral components and broadband noise, the computed amplitude of the DFT output in each bin (filter bank) is significantly affected by the noise contained in the corresponding bandwidth of the filter bank.

*Figure 1: DFT (Discrete Fourier Transform) viewed as processing through a set of filter banks*

*Figure 1: DFT (Discrete Fourier Transform) viewed as processing through a set of filter banks*

In reality, signals are of time-limited nature and nothing can be known about the signal beyond the measured interval. When the time-limited signal slice is subjected to analysis using FFT algorithm (used for computing DFT), the FFT implicitly assumes that the signal essentially repeats itself after the observed interval. This may lead to discontinuities at the edges of each slice, that causes the energy contained in each frequency bin to spill into other bins. This phenomenon is called spectral leakage.

Equivalent Noise Bandwidth:^[2]

To suppress the spectral leakage, the signals are multiplied with a window function so as to smooth the discontinuity at the edges of the FFT slices. As a result, the choice of window function affects the amount of signal and noise that goes inside each filter bank. Hence the amount of noise that gets accumulated into each filter bank is affected by the bandwidth of the chosen window function. In effect, the amplitude estimate in each frequency bin is influenced by the amount of accumulated noise contributed by the window function of choice. In other words, the noise floor displayed at the FFT output, varies according to the window of choice.

Equivalent noise bandwidth (ENBW), often specified in terms of FFT bins, is the bandwidth of a fictitious brick-wall filter such that the amount of noise power accumulated inside the brick-wall filter is same as the noise power accumulated when processing white noise through the selected window.

*Figure 2: Illustrating equivalent noise bandwidth (ENBW) (figure not to scale*)

In discrete domain, the equivalent noise bandwidth B_enbw can be computed using time domain samples as

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

where L is the length of the window.

ENBW is a commonly used metric to characterize the variation in the displayed noise floor if a non rectangular window is used. If noise floor measurements are desired, the ENBW correction factor need to be applied to account for the variation in the noise floor [3]. Table 1 below, lists the ENBW (bins) and ENBW correction factor for some well known window functions.

Following figure illustrates the equivalent noise bandwidth illustrated for Hann window.

*Figure 3: Equivalent noise bandwidth (ENBW) illustrated for Hann window*

Coherent power gain:

In time-domain, windowing a signal reduces the amplitude of the signal at its boundaries (Figure 4). This causes a reduction in the amplitude of the spectral component when FFT used to visualize the signal in frequency domain. This reduction in amplitude in the spectral components is characterized as coherent power gain (which actually is a loss). So, when FFT is computed on a windowed signal, in order to compensate for the loss in the amplitude, we simply add the coherent power gain to the FFT output.

*Figure 4: Windowing a 10 Hz sinusoidal signal with Hann window of length L = 151 (sampling frequency F_s=160 Hz)*

Coherent power gain of a window w[n] of length L is given by

\[\text{Coherent power gain (dB)} = 20 \; log_{10} \left( \frac{\sum_n w[n]}{L} \right)\]

Following plot depicts the coherent power gain (i.e, the reduction in the FFT magnitude when the input signal is processed with a window) of a windowed sinusoidal signal of frequency 10 Hz. The length of the window is L = 151 points and the simulation assumes an oversampling factor of 16 (i.e, F_s=160 Hz). The FFT length is N_FFT =2048.

Table 1 below, lists the coherent power gain for some well known window functions.

*Figure 5: Coherent power gain – illustrated for various window functions*

Scalloping loss

As discussed above, the FFT (equivalently the DFT) operation can be seen as processing a signal through a set of filter banks with bandwidth Δf centered on the successive bin frequencies. The frequency resolution (Δf) of FFT depends on the size of FFT (N_FFT) and the sampling frequency F_s.

\[\Delta f = \frac{F_s}{N_{FFT}}\]

Therefore, the FFT process can measure amplitude of tones that are multiples of the frequency resolution Δf. That is, if the frequency of the observed signal matches a particular bin frequency, the signal amplitude will be maximum at that bin.

However, if the tone of the signal falls between two bins, the signal power is spread between the adjacent bins and hence the displayed signal amplitude/power is reduced.

Scalloping loss quantifies the worst case reduction in the signal level, when the tone of the observed signal falls exactly mid way between two bin frequencies. The worst case scalloping loss is calculated as

\[\text{Scalloping loss} = \frac{\sum_n w[n] e^{\left( -j \frac{\pi}{N_{FFT}}n \right)}}{\sum_n w[n]}\]

In reality, scalloping loss depends on the type of window and the relationship between the signal tones to the bin frequencies. Table 1 below, lists the scalloping loss for some well known window functions.

*Figure 6: Scalloping loss illustrated for processing a signal with Boxcar and Bartlett window*

Figure of merits for window functions

*Table 1: Figure of merits for various window functions*

List of window functions

Boxcar (Rectangular)

\[w[n] = \begin{cases} 1, \quad 0 \leq n \leq L-1 \\ 0, \quad \text{otherwise} \end{cases}\]

Figure 1: Boxcarr (rectangular) window *(L=51)*

Barthann

\[ w[n] = 0.62 – 0.48 \left| \frac{n}{L – 1} – 0.5 \right| + 0.38 cos \left(2 \pi \left| \frac{n}{L – 1} – 0.5 \right| \right), \quad 0 \leq n \leq L-1\]

Bartlett

\[w[n] = \frac{2}{L-1} \left(\frac{L-1}{2} – \left|n – \frac{L-1}{2}\right| \right),\quad 0 \leq n \leq L-1\]

Blackman

\[w[n] = 0.42 – 0.5 \cos(2\pi n/L) + 0.08 \cos(4\pi n/L), \quad 0 \leq n \leq L-1\]

Blackman-Harris

\[w[n]=a_0 – a_1 \cos \left ( \frac{2 \pi n}{L} \right)+ a_2 \cos \left ( \frac{4 \pi n}{L} \right)- a_3 \cos \left ( \frac{6 \pi n}{L} \right), \quad 0 \leq n \leq L-1\] \[a_0=0.35875;\quad a_1=0.48829;\quad a_2=0.14128;\quad a_3=0.01168\]

Bohman

\[w[n] = \left( 1 – |x|\right) cos \left( \pi |x| \right) + \frac{1}{\pi} sin \left( \pi |x| \right) , \quad -1 \leq x \leq 1 \]

Cosine

\[w[n] = sin \left( \frac{\pi}{ L } \left(n + 0.5 \right) \right),\quad 0 \leq n \leq L-1\]

Flattop

\[w[n]=a_0 – a_1 \cos \left ( \frac{2 \pi n}{L} \right)+ a_2 \cos \left ( \frac{4 \pi n}{L} \right)- a_3 \cos \left ( \frac{6 \pi n}{L} \right), \quad 0 \leq n \leq L-1 \\ a_0=0.21557895;\; a_1=0.41663158 \\ a_2=0.277263158;\; a_3=0.006947368 \]

Hamming

\[w[n] = 0.54 – 0.46 \cos\left(\frac{2\pi{n}}{L-1}\right), \quad 0 \leq n \leq L-1\]

Hann

\[w[n] = 0.5 – 0.5 \cos\left(\frac{2\pi{n}}{L-1}\right), \quad 0 \leq n \leq L-1\]

Nuttall

\[ w[n]=a_0 – a_1 \cos \left ( \frac{2 \pi n}{L} \right)+ a_2 \cos \left ( \frac{4 \pi n}{L} \right)- a_3 \cos \left ( \frac{6 \pi n}{L} \right), \quad 0 \leq n \leq L-1 \\ a_0=0.3635819;\; a_1=0.4891775;\\ a_2=0.1365995;\; a_3=0.0106411\]

Parzen

\[w[n] = \begin{cases} 1- 6 \left( \frac{|n|}{L/2} \right)^2 + 6 \left( \frac{|n|}{L/2} \right)^3, & 0 \leq |n| \leq (L-1)/4 \\ 2 \left( 1 – \frac{|n|}{L/2} \right)^3, & (L-1)/4 < |n| \leq (L-1)/2 \end{cases}\]

Triangular

\[w[n] = 1 – \frac{|n|}{L/2}, \quad n = -L/2, \cdots, -1, 0, 1, \cdots, L/2\]

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References:

[1] Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform”, proceedings of IEEE, vol. 66, no. 1, January 1978 .↗
[2] Stefan Scholl, “Exact Signal Measurements using FFT Analysis”,Microelectronic Systems Design Research Group, TU Kaiserslautern, Germany.↗

Additional Resources:

[1] If you are particularly interested in calculating the Equivalent Noise Bandwidth of Analog Filters refer – Thomas J. Karras,”Equivalent Noise Bandwidth Analysis from Transfer Functions”,NASA Technical Note,NASA TN D-2842.↗
[2] Strategies for Choosing Windows : Michael Cerna and Audrey F. Harvey,”The Fundamentals of FFT-Based Signal Analysis and Measurement”,National Instruments Application Notes 041.↗

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Key focus: Equivalent noise bandwidth (ENBW), is the bandwidth of a fictitious brick-wall filter that allows same amount of noise as a window function. Learn how to calculate ENBW in applications involving window functions and FFT operation.

FFT and spectral leakage

Here is an article with illustrations on – FFT slices and the phenomenon of spectral leakage,

Hence, to suppress the spectral leakage, the signals are multiplied with a window function so as to smooth the discontinuity at the edges of the FFT slices. As a result, the choice of window function affects the amount of signal and noise that goes inside each filter bank. Hence the amount of noise that gets accumulated into each filter bank is affected by the bandwidth of the chosen window function. In effect, the amplitude estimate in each frequency bin is influenced by the amount of accumulated noise contributed by the window function of choice. In other words, the noise floor displayed at the FFT output, varies according to the window of choice.

Equivalent noise bandwidth

In other words, in frequency domain, the power accumulated in the chosen window is given by

\[P_{window} = \int_{-f}^{f} |W(f)|^2 df\]

Then we wish to find the equivalent noise bandwidth B_enbw, such that

\[P_{brick-wall} = B_{enbw} |W(f_0)|^2 = P_{window}\]

where f₀ is the frequency at which the power peaks and P_brick-wall is the power accumulated in the fictitious rectangular window of bandwidth B_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, k₀ is the index at which the magnitude of FFT output is maximum and N is the window length. Applying Parseval’s theorem and with a window of length L, B_enbw can also be computed using time domain samples as

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

Following figure illustrates the equivalent noise bandwidth illustrated for Hann window.

Python script

ENBW can be calculated from the time domain samples in a straightforward manner. Following Python 3.0 script calculates the ENBW for some well-known window functions provided as part of Scipy module↗.

#Author : Mathuranathan Viswanathan for gaussianwaves.com
#Date: 13 Sept 2020
#Script to calculate equivalent noise bandwidth (ENBW) for some well known window functions

import numpy as np
import pandas as pd
from scipy import signal

def equivalent_noise_bandwidth(window):
    #Returns the Equivalent Noise BandWidth (ENBW)
    return len(window) * np.sum(window**2) / np.sum(window)**2

def get_enbw_windows():
    #Return ENBW for all the following windows as a dataframe
    window_names = ['boxcar','barthann','bartlett','blackman','blackmanharris','bohman','cosine','exponential','flattop','hamming','hann','nuttall','parzen','triang']
    
    df = pd.DataFrame(columns=['Window','ENBW (bins)','ENBW correction (dB)'])
    for window_name in window_names:
        method_name = window_name
        func_to_run = getattr(signal, method_name) #map window names to window functions in scipy package
        L = 16384 #Number of points in the output window
        window = func_to_run(L) #call the functions
        
        enbw = equivalent_noise_bandwidth(window) #compute ENBW
        
        df = df.append({'Window': window_name.title(),'ENBW (bins)':round(enbw,3),'ENBW correction (dB)': round(10*np.log10(enbw),3)},ignore_index=True)
                       
    return df

df = get_enbw_windows() #call the function
df.head(14) #display dataframe

The resulting output is displayed in the following table (dataframe)

*Table 1: Table of equivalent noise bandwidth for some well known window functions*

As an example, the following plot depicts the difference in the noise floor of FFT output of a noise added sine wave that is processed through Boxcar and Flattop window.

*Figure 4: Noise floor and ENBW for flattop & boxcar window (FFT output) for noise added 10 Hz sinewave (oversampling factor = 16, Fs = 160 Hz, window length L =2048, SNR = 30 dB)*

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Reference

[1] Stefan Scholl, “Exact Signal Measurements using FFT Analysis”,Microelectronic Systems Design Research Group, TU Kaiserslautern, Germany.↗