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Unformatted text preview: 1 Chapter 6: Discrete Fourier Transform & Fast Fourier Transform Discrete Fourier Transform (DFT) Introduction DFT and its Properties Circular Convolution Linear Convolution Using DFT Fast Fourier Transform (FFT) Decimation in Time Decimation in Frequency TC Chuah, Haris, April 2007, FOE, MMU ETM4096 Digital Signal Processing 2 Types of Fourier Transform A signal can be continuous or discrete , and periodic or aperiodic . Fourier transform can be categorised based on the signal type : It extends into positive and negative infinity without repeating in a periodic pattern e.g. y = e x and y = x 2 . The Fourier transform of such signal : Fourier Transform . It repeats itself in a regular pattern between positive and negative infinity e.g. y = sin( x ) and square waves. The Fourier transform of such signal : Fourier Series . AperiodicContinuous PeriodicContinuous 3 Defined at discrete points between positive and negative infinity not repeating in a periodic pattern. The Fourier transform of such signal : Discrete Time Fourier Transform ( DTFT ). Discrete signal that repeats itself in a regular pattern between positive and negative infinity . The Fourier transform of such signal : Discrete Fourier Transform (DFT). The four classes of signal above extend to positive and negative infinity. How about signals with finite length ? AperiodicDiscrete PeriodicDiscrete Types of Fourier Transform 4 1. Assume that the finite length signal extends infinitely to positive and negative with zero values . The signal will look aperiodic and discrete. Thus DTFT can be applied. Types of Fourier Transform ] [ n x  n ] [ n x  n 1. Assume that the signal repeats itself to positive and negative infinitely . The signal will look periodic and discrete. Thus DFT can be applied. There are two ways to get around this: 5 It turns out that an infinite number of sinusoids are required to synthesise a signal that is aperiodic. This makes it impossible to calculate DTFT of such signal in computer algorithm . Thus the only type of Fourier transform that can be used in DSP is thus the DFT . Unlike the DTFT, which is a continuous function of a continuous variable, , as will be seen later, the DFT is a sequence that corresponds to samples of the DTFT . Such a representation is very useful for digital computation and for digital hardware implementation. Types of Fourier Transform ( 29  = = k jk j e k x e X ] [ From Chapter 1, the DTFT of a sequence is defined as: 6 Fourier Series: Continuous in Time, Discreet in Frequency Fourier series says that we can synthesize a periodic waveform by using a weighted sum of harmonically related sinusoid. The weight in the formula is a k....
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