Chapter 4 Discrete-Time Signals and transform

Discrete-time signal Discrete-time signal Sampled signal using sampling period T where is signal value at n Fig. 4-1.

Discrete Fourier transform Continuous Fourier Transform Discrete signal using impulse train (4-1) (4-2) where s(t) is impulse train with period T.

Discrete Fourier transform of discrete-time signal (4-3) where x(nT)=xn .

Properties of discrete Fourier transform If x(t) is even function, that is xn= x-n , is real and even function. Inverse is valid. If x(t) is odd function, that is xn= -x-n , is imaginary and odd function. Inverse is valid. and are complex conjugate. is periodic signal at period of (4-4)

Calculation of discrete Fourier transform (4-5) where N is the number of samples in a period for periodic signal or in data window for random signal.

Calculation of discrete spectral values N samples of have independent values In Eq.(4-5), is replaced by Properties of discrete Fourier transform (4-6) (4-7) (4-8) (4-9) (4-10)

Inverse discrete Fourier transform Proof (4-11) (4-12) where (4-13)

Discrete Fourier Transform pair (4-14) (4-15) where , and N and T represent the numbers of samples and sampling interval, respectively

Example 4-1 Calculate DFT for discrete sequence as {1,0,0,1} m=0, m=1,

Example 4-2 Calculate DFT and spectrum for discrete-time signal in Fig. 4-2 Fig. 4-2.

Fourier transform for n = 4

Calculation of discrete frequency using Eq. (4-6) Spectrum Nyquist frequency Fig. 4-3.

Relation with Fourier transforms Relations between and Sampled signal xs(t) is given by (4-16) (4-17) where and (4-18) (4-19)

Fourier transform of xs(t) Then Fourier transform of xs(t) (4-20) (4-21) (4-22)

Relationship between Fourier transform and discrete Fourier transform Fourier transform of x(t) Discrete Fourier transform for T = T1 Discrete Fourier transform for T = T2(T 2 = 2T1) Fig. 4-4.

Truncation and spectrum leakage Truncation for x(t) Samples of x(t) = 2e-t Amplitude spectrum of x(t) Amplitude spectrum of truncated x(t) with NT = 2[sec] Fig. 4-5.

Effect of truncation Fourier transform for truncated x(t) with t = NT (4-23) (4-24) where is truncation error and is shown in Fig. 4-5(b) as ripple.

Analysis of truncation effect Comparison between time and frequency domains x(t) and its amplitude spectrum Fig. 4-6.

Rectangle function w(t) with NT = 2 and its spectrum Multiplication between w(t) and x(t) and its spectrum Fig. 4-6.

Role of window function to reduce spectrum leakage Windowing Spectrum leakage occurs due to truncation Reduce the spectral leakage using various window functions Hanning window Hamming window Blackman window, etc

Spectrum of window function Table. 4-1. Window type Window function Spectrum of window function Rectangular Bartlett Hanning Hamming Papoulis Blackman Parzen

Hanning window Hanning window in discrete-time (4-25) (4-26)

Fourier transform of Hanning window Spectrum of Hanning window for (4-27) (4-28) Fig. 4-8.

Comparison between Hanning and rectangular window Reduction of spectral leakage using Hanning window rectangle window Hanning window Fig. 4-7.

Discrete convolution Discrete convolution Discrete convolution in time domain where is periodic sample set as (4-29) (4-30) (4-31)

Proof by using inverse discrete Fourier transform (4-32) (4-33) where (4-34) (4-35)

Discrete convolution Sample sets xm and hn-m for periodic convolution Fig. 4-9.

Discrete convolution for N=4 Using periodic property gives h-1=h3, h-2=h2, and h-3=h1

Discrete convolution using zero padding Input samples Periodic samples hn using zero padding Fig. 4-10.

Example 4-3 Analog convolution Discrete (periodic) convolution Discrete convolution using zero padding

Discrete convolution in frequency domain Proof (4-36) (4-37) (4-38)

Fast Fourier transform Redundancy parts of DFT Discrete Fourier transform Redundancy part (4-39) where m is constant for xn is n-th sample for x(t), and N is number of samples. (4-40)

Discrete Fourier transform (4-41) where Fig. 4-11.

Discrete Fourier transform for N=8 Separation of even and odd terms (4-42) (4-43) where (4-44) (4-45)

Discrete Fourier transform for N-samples , even and odd terms Discrete Fourier transform for N-samples (4-46) (4-47)

Fast Fourier transform in time domain FFT for N=8 Separation of even and odd terms (4-48) (4-49) (4-50)

Even terms Property of (4-51) (4-52) Periodic property (4-53)

Property of Periodic property (4-54)

Property of Periodic property (4-55)

Property of Property of and Periodic property Periodic property (4-56) (4-57) (4-58)

Signal-flow graph for N=2 and N=8 Fig. 4-12. Fig. 4-13.

Decomposition Fig. 4-14.

Decomposition using bit reversal Fig. 4-15.

Computational complexity of FFT Number of complex multiplication Ratio of computational complexity between DFT and FFT (4-59) (4-60) Table. 4-2. N Discrete Fourier transform Fast Fourier transform 2 4 1 8 64 12 32 1024 80 4096 192 512 262144 2304 1048576 5120