CH07 Frequency Domain Processing

CH07 Frequency Domain Processing - CHAPTER 7 Frequency...

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C H A P T E R 7 Frequency Domain Processing Transformation of signals to the frequency domain is widely used in signal processing. In many cases, such transformations provide a more effective repre- sentation and a more computationally efficient processing of signals as compared to time domain processing. For example, due to the equivalency of convolution operation in the time domain to multiplication in the frequency domain, one can find the output of a linear system by simply multiplying the Fourier transform of the input signal by the system transfer function. This chapter presents an overview of three widely used frequency domain trans- formations, namely fast Fourier transform (FFT), short-time Fourier transform (STFT), and discrete wavelet transform (DWT). More theoretical details regarding these transformations can be found in many signal processing textbooks, e.g. [ 1 ]. 7.1 Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) Discrete Fourier transform (DFT) X ½ k of an N -point signal x ½ n is given by X ½ k ¼ X N ± 1 n ¼ 0 x ½ n W nk N ; k ¼ 0 ; 1 ; ::: ; N ± 1 x ½ n ¼ 1 N X N ± 1 n ¼ 0 X ½ k W ± nk N ; n ¼ 0 ; 1 ; ::: ; N ± 1 8 > > > > > > < > > > > > > : (7.1) where W N ¼ e ± j 2 p = N . The above transform equations require N complex multiplications and N ± 1 complex additions for each term. For all N terms, N 2 complex multiplications and N 2 ± N complex additions are needed. As it is well known, the direct computation of (7.1) is not efficient. 175
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To obtain a fast or real-time implementation of (7.1) , one often uses a fast Fourier transform (FFT) algorithm, which makes use of the symmetry properties of DFT. There are many approaches to finding a fast implementation of DFT; that is, there are many variations of FFT algorithms. Here, we mention the approach presented in the TI Application Report SPRA291 for computing a 2 N -point FFT [ 2 ]. This approach involves forming two new N -point signals x 1 ½ n and x 2 ½ n from a 2 N -point signal g ½ n by splitting it into an even and an odd part as follows: x 1 ½ n ¼ g ½ 2 n 0 ² n ² N ± 1 x 2 ½ n ¼ g ½ 2 n þ 1 (7.2) From the two sequences x 1 ½ n and x 2 ½ n , a new complex sequence x ½ n is defined to be x ½ n ¼ x 1 ½ n þ jx 2 ½ n 0 ² n ² N ± 1 (7.3) To get G ½ k , the DFT of g ½ n , the equation G ½ k ¼ X ½ k A ½ k þ X ³ ½ N ± k B ½ k k ¼ 0 ; 1 ; ::: ; N ± 1 ; with X ½ N ¼ X ½ 0 (7.4) is used, where A ½ k ¼ 1 2 ð 1 ± jW k 2 N Þ (7.5) and B ½ k ¼ 1 2 ð 1 þ jW k 2 N Þ (7.6) Only N points of G ½ k are computed from (7.4) . The remaining points are found by using the complex conjugate property of G ½ k , that is, G ½ 2 N ± k ¼ G ³ ½ k . As a result, a 2 N -point transform is calculated based on an N -point transform, leading to a reduction in the number of operations. 7.2 Short-Time Fourier Transform (STFT) Short-time Fourier transform (STFT) is a sequence of Fourier transforms of a windowed signal. STFT provides the time-localized frequency information for situations in which frequency components of a signal vary over time, whereas the
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