Is an endless succession of replicas of the spectrum

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is an endless succession of replicas of the spectrum of the original signal, spaced by intervals of the sampling frequency ω = 2 π/T . The replicas can be removed easily by low-pass filtering. None of the original information is lost so long as the replicas do not overlap or alias onto one another. Overlap can be avoided so long as the sampling frequency is as great or greater than the total bandwidth of the signal, a number bounded by twice the maximum frequency component. This requirement defines the Nyquist sampling frequency ω s = 2 | ω | max . In practice, ideal low-pass filtering is impossible to implement, and oversampling at a rate above the Nyquist frequency is usually performed. ω Y( ω29 B 2 π/ T Figure 1.5: Illustration of the sampling theorem. The spectrum of the sampled signal is an endless succession of replicas of the original spectrum. To avoid frequency aliasing, the sampling theorem requires that B 2 π/T . 18
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Discrete Fourier Transform The sampling theorem implies a transformation between a discrete set of samples and a discrete set of Fourier ampli- tudes. The discrete Fourier transform (DFT) is defined by F m = N 1 summationdisplay n =0 f n e j 2 πnm/N , m = 0 , · · · ,N 1 f n = 1 N N 1 summationdisplay m =0 F m e j 2 πnm/N , n = 0 , · · · ,N 1 where the sequences F m and f n can be viewed as frequency- and time- domain representations of a periodic, band- limited sequence with period N . The two expressions follow immediately from the Fourier series formalism outlined earlier in this section, given discrete time sampling. Expedient computational means of evaluating the DFT, notably the fast Fourier transform or FFT, have existed for decades and are at the heart of many signal processing and display numerical codes. Most often, N is made to be a power of 2, although this is not strictly necessary. DFTs and FFTs in particular are in widespread use wherever spectral analysis is being performed. Offsetting their simplicity and efficiency is a serious pathology arising from the fact that time series in a great many real-world situations are not periodic. Power spectra computed using periodograms (algorithms for performing DFTs that assume the periodicity of the signal) suffer from artifacts as a result, and these may be severe and misleading. No choice of sample rate or sample length N can completely remedy the problem. The jump between the first and last sample in the sequential time series will function like a discontinuity upon analysis with a DFT, and artificial ringing will appear in the spectrum as a result. A non-periodic signal can be viewed as being excised from longer, periodic signal by means of multiplication by a gate function. The Fourier transform of a gate function is a sinc function. Consequently, the DFT spectrum reflects something like the convolution of the true F ( ω ) with a sinc function, producing both ringing and broadening. Artificial sidelobes down about -13.5 dB from the main lobe will accompany features in the spectrum, and these may be mistaken for actual frequency components of the signal.
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