Side Lobes

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Spectral Audio Signal Processing

The Rectangular Window

Summary

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A periodic signal is a signal that forever repeats itself. — Click for http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals

A Fourier Series (FS) expansion is the appropriate Fourier transform for periodic signals. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Series_FS_Relation.html

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The aliased sinc function is obtained from the sinc function by applying the aliasing operator. It arises in digital audio as the spectrum of a sampled rectangular pulse. — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window.html

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

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Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

The Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

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The 'Cardinal sine function' is defined as sin(pi x)/(pi x) — Click for https://ccrma.stanford.edu/~jos/mdft/Sinc_Function.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

The Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.html

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The 'Cardinal sine function' is defined as sin(pi x)/(pi x) — Click for https://ccrma.stanford.edu/~jos/mdft/Sinc_Function.html

The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Time_Fourier_Transform.html

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Spectral Audio Signal Processing

The Rectangular Window

Summary

Rectangular Window Side-Lobes

From Fig.3.3 and Eq.(3.4), we see that the main-lobe width is $2\cdot 2\pi/M=4\pi/11 \approx 1.1$ radian, and the side-lobe level is 13 dB down.

Since the DTFT of the rectangular window approximates the sinc function (see (3.4)), which has an amplitude envelope proportional to $1/\omega$ (see (3.7)), it should ``roll off'' at approximately 6 dB per octave (since $-20\log_{10}(2)=6.0205999\ldots$ ). This is verified in the log-log plot of Fig.3.6.

**Figure 3.6:** Roll-off of the rectangular-window Fourier transform.
$\includegraphics[width=\textwidth ]{eps/rectWindowLLFT}$

As the sampling rate approaches infinity, the rectangular window transform ( $\hbox{asinc}$ ) converges exactly to the sinc function. Therefore, the departure of the roll-off from that of the sinc function can be ascribed to aliasing in the frequency domain, due to sampling in the time domain (hence the name `` $\hbox{asinc}$ '').

Note that each side lobe has width $\Omega_M \isdeftext 2\pi/M$ , as measured between zero crossings.^4.3 The main lobe, on the other hand, is width $2\Omega_M$ . Thus, in principle, we should never confuse side-lobe peaks with main-lobe peaks, because a peak must be at least $2\Omega_M$ wide in order to be considered ``real''. However, in complicated real-world scenarios, side-lobes can still cause estimation errors (``bias''). Furthermore, two sinusoids at closely spaced frequencies and opposite phase can partially cancel each other's main lobes, making them appear to be narrower than $2\Omega_M$ .

In summary, the DTFT of the -sample rectangular window is proportional to the `aliased sinc function':

$\begin{eqnarray*} \hbox{asinc}_M(\omega) &\isdef & \frac{\sin(\omega M / 2)}{M\cdot\sin(\omega/2)} \\ [0.2in] &\approx& \frac{\sin(\pi fM)}{M\pi f} \isdefs \mbox{sinc}(fM) \end{eqnarray*}$

Thus, it has zero crossings at integer multiples of

$\displaystyle \Omega_M \isdefs \frac{2\pi}{M}.$

(4.11)

Its main-lobe width is $2\Omega_M$ and its first side-lobe is 13 dB down from the main-lobe peak. As

gets bigger, the main-lobe narrows, giving better frequency resolution (as discussed in the next section). Note that the window-length

has no effect on side-lobe level (ignoring aliasing). The side-lobe height is instead a result of the abruptness of the window's transition from 1 to 0 in the time domain. This is the same thing as the so-called Gibbs phenomenon seen in truncated Fourier series expansions of periodic waveforms. The abruptness of the window discontinuity in the time domain is also what determines the side-lobe roll-off rate (approximately 6 dB per octave). The relation of roll-off rate to the smoothness of the window at its endpoints is discussed in §B.18.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Rectangular Window Side-Lobes

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University