Define the Problem for the Spectral Analysis Case Study

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A basic digital spectrum analyzer is shown in the figure. The default window, w[n], is a constant of one over the capture interval of Nr samples. The FFT works with a finite length discrete-time signal. The window function w[n] is a design parameter that you may consider changing later in the process. The antialising filter ensures that signals greater than fs/2 don’t enter the ADC.

[Credit: Illustration by Mark Wickert, PhD]
Credit: Illustration by Mark Wickert, PhD

The signal model for this study is


where wn(t) represents noise, which adds another dimension to the problem and ultimately leads to spectral estimation for random signals. The noise-free model serves as a good starting point. You can assume that the sinusoids of r(t) are spectrally contained in the Nyquist band [0, fs / 2], which means that the antialiasing filter imparts no distortion to the signal of interest.

The study of the DTFT reveals that the Fourier transform of rw[n] = r[n] x w[n] is given by the following integral:


This integral is a periodic convolution in the frequency domain — periodic because the functions involved have period 2π. The big deal is that the convolution operation tends to spread things out (except when one function is an impulse). Spreading the spectrum is bad.

Spectral content that starts out focused in an area, or at one frequency, is now spread to a much wider interval on the frequency axis. Spectral spreading, or leakage, can cover up other spectral content of interest and make it difficult to discern two closely spaced signals. But not using a window isn’t an option because a finite data record has a starting point and an ending point, which by itself defines a rectangular window.

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