 This series is excerpted from "Digital Signal Processing
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Part 1 explains the DFT and FFT. Part 3 covers parametric estimation and wavelets. It
will be published Thursday, March 6.
Spectral analysis, by estimation of the power spectrum or spectral power density of a deterministic or random signal, involves performing a squaring function. Obtaining a good estimate of the spectrum, i.e. the signal power contents as a function of the frequency, is not entirely easy in practice. The main problem is that in most cases we only have access to a limited set of samples of the signal; in another situation, we are forced to limit the number of samples in order to be able to perform the calculations in a reasonable time. These limitations introduce errors in the estimate. If the signal is a randomtype signal, we may also obtain large fluctuations in power estimates based on samples from different populations. Hence, in the general case there is no way to obtain the true power spectrum of a signal unless we are allowed to observe it infinitely. This is why the term estimation is frequently used in this context.
5.2.1 Discrete Fourier transform and fast Fourier transform approaches
Spectral analysis using the Fourier transform, a non-parametric method, was originally proposed in 1807 by the French mathematician and Baron J.B.J. Fourier. The discrete version of this transform, commonly used today in digital signal processing (DSP) applications, is called discrete Fourier transform (DFT). A smart algorithm for calculating the DFT, causing less computational load for a digital computer, is the fast Fourier transform (FFT). From a purely mathematical point of view, DFT and FFT do the same job as shown above. Considering a sampled signal x(n) for −∞ < n < ∞ further, assume that the signal has finite energy, i.e.
As seen above, the DFT of the signal can then be expressed as
where ωs is the sampling frequency. By Parseval's relation, the energy of the signal can then be expressed as
where Sxx(ω) is referred to as the spectral density of the signal x(n). The spectral density is hence the squared magnitude of the Fourier transform of the signal. As mentioned above, in most cases the signal is not defined for −∞ <n < ∞, or we may not be able to handle an infinite length sequence in a practical application. In such a case, the signal is assumed to be non-zero only for n = 0, 1, ..., N − 1 and assumed to be zero otherwise. The spectral density is then written as
The power of the signal as a function of the frequency is called a periodogram, which A. Schuster defined in 1898 as
In the case that the signal x(n) exists over the entire interval (−∞,∞) and its energy is infinite (e.g. sinusoidal or other periodic signals), it is convenient to define the spectral density as (Mitra and Kaiser, 1993)
Now, in many practical cases, to use an FFT algorithm, N is chosen to be a power of 2 (e.g. 64, 128, 256, …). This means that the spectral density can only be calculated at N discrete points, which in many cases turns out to be too sparse. Quite often, we need a finer frequency spacing, i.e. we need to know the spectrum at L points where L > N . This can be accomplished by zero padding, so that the data sequence consisting of N samples is extended by adding L − N zero value samples to the end of the sequence. The "new" L point sequence is then transformed using DFT or FFT. The effective frequency spacing is now
It is important to remember that zero padding does not increase the true resolution in frequency, it merely interpolates the spectrum at more points. However, for many applications, this is sufficient.
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