1. Chapter 4 Discrete-Time Signals and transform

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

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

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

5. 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)

6. 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.

7. 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)

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

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

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

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

12. Fourier transform for n = 4

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

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

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

16. 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.

17. 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.

18. 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.

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

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

21. 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

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

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

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

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

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

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

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

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

30. Discrete convolution using zero padding
Input samples
Periodic samples hn using zero padding
Fig

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

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

33. 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)

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

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

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

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

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

39. Property of
Periodic property
(4-54)

40. Property of
Periodic property
(4-55)

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

42. Signal-flow graph for N=2 and N=8
Fig
Fig

43. Decomposition
Fig

44. Decomposition using bit reversal
Fig

45. 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
5120