Discrete Fourier Transform (DFT)

1. Discrete Fourier Transform (DFT) Hafiz Malik Dept. of Electrical & Computer Engineering The University of Michigan-Dearborn hafiz@umich.edu http://www-perosnal-engin.umd.umich.edu/~hafiz DFT - Hafiz Malik

2. Acknowledgement • These lecture slides are based on • Chapter 5 of • Digital Signal Processing: Principles, Algorithms and Applications 3rd e • John G. Proakis and Dimitris K Manolakis • Lecture slides of Prof. Richard M. Stern • Department of Electrical and Computer Engineering, Carnegie Mellon University DFT - Hafiz Malik

3. Frequency Domain Sampling • Consider an aperiodic signal finite duration signal with FT • Suppose we sample periodically in frequency at spacing radians between successive samples. • Since is periodic with period , therefore only samples in the fundamental frequency range are required. • Let we take N equidistant samples  DFT - Hafiz Malik

4. Frequency Domain Sampling • Now calculate Eq. (1) at , i.e., • Summation in Eq. (2) can be written as DFT - Hafiz Malik

5. Frequency Domain Sampling • If we change the index in the inner summation from n n – lN & interchanging the order of summation, we have, For k = 0, 1, … , N – 1 • Here, signal is a periodic sequence with fundamental period N DFT - Hafiz Malik

6. Frequency Domain Sampling • The xp(n) can be expresses using FS as, where Fourier coefficient ck is given as, By comparing Eq. (4) & (6) we observe, DFT - Hafiz Malik

7. Frequency Domain Sampling • The xp(n) can be expresses using FS as, where Fourier coefficient ck is given as, By comparing Eq. (4) & (6) we observe, Therefore, DFT - Hafiz Malik

8. Frequency Domain Sampling • Here Eq. (8) provides a relationship for reconstruction of the periodic signal xp(n) from the samples of spectrum X(ω). • However, it does not imply that we can recover X(ω) or x(n) from samples. • To determine this we need to consider relationship between xp(n) and x(n). DFT - Hafiz Malik

9. Relationship between xp(n) & x(n) • Since xp(n) is the periodic extension of x(n)  x(n) can be recovered from xp(n) if there is no aliasing in time domain. • To illustrate this fact consider a finite duration sequence x(n) DFT - Hafiz Malik

10. Illustration of Time Domain Aliasing n L L < N xp(n) x(n) L N n xp(n) L > N N DFT - Hafiz Malik

11. Relationship between xp(n) & x(n) • It can be observed from previous example that • From N >= L, x(n) can be recovered from xp(n). • Whereas, N < L, , x(n) cannot be recovered from its periodic extensionxp(n) due to time-domain aliasing • Therefore, X(ω) of an apriodic discrete signal with finite duration L, can be exactly recovered from its samples at frequencies ωk= 2πk/N, if N >= L DFT - Hafiz Malik

12. Discrete Fourier Transform DFT - Hafiz Malik

13. Discrete Fourier Transforms • Consider a sequence x(n) of finite duration L<=N • Then xp(n) is simply a periodic repetition of x(n) where xp(n) over a single period can be expressed as, DFT - Hafiz Malik

14. Discrete Fourier Transforms • Recall Eq. (8) which states that frequency samples can be used to reconstruct x(n). DFT - Hafiz Malik

15. Discrete Fourier Transforms • Therefore, a finite duration sequence x(n) has a FT • The samples of X(ω) at equally spaced frequencies k = 0,1, ... , N-1 where L<=N are given as, • This is the forward discrete Fourier Transform (DFT) DFT - Hafiz Malik

16. Discrete Fourier Transforms • The DFT is periodic in frequency k i.e. • This is as expected since the spectrum is periodic in frequency DFT - Hafiz Malik

17. Inverse Discrete Fourier Transforms • Multiply both sides of the DFT by • And add over the frequency index k • From which DFT - Hafiz Malik

18. Discrete Fourier Transforms • This is the inverse DFT DFT - Hafiz Malik

19. Few Observations • The DFT assumes that we deal with periodic signals in the time domain • Sampling in one domain produces periodic behaviour in the other domain DFT - Hafiz Malik

20. DFT as a Linear Transformation • The forward and inverse DFT can be expressed as: • The DFT • IDFT • where which is Nth root of unity. DFT - Hafiz Malik

21. DFT as a Linear Transformation • Let us define • an N-point vector xNof the sequence x(n), n =0,1,…,N – 1, • an N-point vector XNof frequency samples, and DFT - Hafiz Malik

22. DFT as a Linear Transformation • an N x N matrix WN as Where WN exhibits the following properties Where denotes the complex conjugate of the matrix WN DFT - Hafiz Malik

23. DFT as a Linear Transformation • The N-point DFT using matrix notation can be expressed as, • If we assume that the inverse of WN exists, then Eq. (22) can be inverted by premultiplying both sides by WN -1 results, • This expression is IDFT if we scale RHS by 1/N, i.e., DFT - Hafiz Malik

24. Example DFT - Hafiz Malik

25. Properties of DFT • Periodicity • If x(n) and X(k) are an N-point DFT pair, then • Proof: DFT - Hafiz Malik

26. Properties of DFT • Linearity • If • and • Then for any real- or complex–valued constants a1 and a2 • Proof: DFT - Hafiz Malik

27. Properties of DFT • Circular Symmetry • Circular shift of a sequence x(n) can be represented as the index modulo N, which can be expressed as, • An N-point sequence is called circularly even if it is symmetric about point zero on the circle  • An N-point sequence is called circularly odd if it is antisymmetric about point zero on the circle  • Time reversal of an N-point sequence is attained by reversing its samples about point zero on the circle, i.e., DFT - Hafiz Malik

28. Symmetric Properties of DFT • Let us assume that the N-point sequence x(n) and its DFT are both complex valued, i.e., • TheN-point DFT of x(n) can be expresses as, • TheN-point IDFT of X(k) can be expresses as, DFT - Hafiz Malik

29. Symmetric Properties of DFT • Real-valued Sequence • If sequence x(n) is real then, • Consequently, DFT - Hafiz Malik

30. Symmetric Properties of DFT • Real and Even Sequence • If sequence x(n) is real and even, i.e. • Then Eq. (33) yields XI (k) = 0, and DFT reduces to • which is a real-valued and even. In addition, as XI (k) = 0, the IDFT reduces to DFT - Hafiz Malik

31. Symmetric Properties of DFT • Real and Odd Sequence • If sequence x(n) is real and odd, i.e. • Then Eq. (32) yields XR(k) = 0, then the DFT reduces to • which is a purely imaginary and odd. In addition, as XR (k) = 0, the IDFT reduces to DFT - Hafiz Malik

32. Symmetric Properties of DFT • Purely Imaginary Sequence • If sequence x(n) is imaginary, i.e. • Then Eq. (32) and (33) reduce to • It can be observed from Eq. (41) and (42) that as XR (k) is odd and XI (k) is even. • Furthermore, • if x(n) odd then XI (k) = 0  X(k) is real. • And if x(n) even then XR (k) = 0  X(k) is purely imaginary. DFT - Hafiz Malik

33. Symmetric Properties of DFT • Summary of symmetry properties of DFT • The symmetry properties can be summarized as, DFT - Hafiz Malik

34. Multiplication of two DFT Sequences: Convolution in DFT • Consider the following transform pairs • Define • Find 34 DFT - Hafiz Malik

35. Convolution in DFT • From IDFT • However 35 DFT - Hafiz Malik

36. Convolution in DFT • Or • Thus • This the Circular Convolution 36 DFT - Hafiz Malik

37. Computational Complexity of DFT • It can be observed from Eq. (10) that the computation of each DFT point, X(k), requires N complex multiplications, and N – 1 complex additions. • Therefore, the N-point DFT computation requires N2 complex multiplications, and N(N – 1) complex additions  O(N 2) is the complexity of DFT. DFT - Hafiz Malik