UNIT-III Discrete and Fast Fourier Transform

1. Digital Signal Processing SSGMCE Shegaon UNIT-IIIDiscrete and Fast Fourier Transform Prof. V. N. Bhonge Dept. of Electronics & Telecomm. Shegaon – 444203 vnbhonge@gmail.com

2. SSGMCE Shegaon UNIT-IIIDiscrete and Fast Fourier Transform Discrete and Fast Fourier Transform: Discrete convolution, Discrete Time FourierTransforms (DTFT), Fast Fourier Transform (FFT), Computing an Inverse DFT by Doing a Direct DFT. Fast convolution, correlation. (9). Prof. V. N. Bhonge Dept. of E & T

3. Discrete-Time Fourier Transform SSGMCE Shegaon Discrete-Time Fourier Transform • Notes: • The DTFT and inverse DTFT are not symmetric. One is integration over a finite interval (2π), and the other is summation over infinite terms. • The signal, x[n] is aperiodic, and hence, the transform is a continuous function of frequency. (Recall, periodic signals have a line spectrum.) • The DTFT is periodic with period 2. Later we will exploit this property to develop a faster way to compute this transform. Prof. V. N. Bhonge Dept. of E & T

4. Example 1: 1st Order System, Decay Power stable system Calculate the DT Fourier transform of the signal: • Therefore: a=0.8

5. Example 2: Rectangular Pulse N1=2 • Consider the rectangular pulse • and the Fourier transform is

6. Example 3: Impulse Signal • Fourier transform of the DT impulse signal is

7. Properties: Periodicity, Linearity & Time • The DT Fourier transform is always periodic with period 2p, because X(ej(w+2p)) = X(ejw) • It is relatively straightforward to prove that the DT Fourier transform is linear, i.e. • Similarly, if a DT signal is shifted by n0 units of time

8. Convolution in the Frequency Domain • Like continuous time signals and systems, the time-domain convolution of two discrete time signals can be represented as the multiplication of the Fourier transforms • If x[n], h[n] and y[n] are the input, impulse response and output of a discrete-time LTI system so, by convolution, y[n] = x[n]*h[n] • Then Y(ejw) = X(ejw)H(ejw) • The proof is analogous to proof used for the convolution of continuous time Fourier transforms • Convolution in the discrete time domain is replaced by multiplication in the frequency domain.

9. Example: 1st Order System • Consider an LTI system with impulse response h[n]=anu[n], |a|<1 • and the system input is x[n]=bnu[n], |b|<1 • The DT Fourier transforms are: So • Expressing as partial fractions, assuming ab: • and spotting the inverse Fourier transform

10. Lecture 11: Summary • Apart from a slightly difference, the Fourier transform of a discrete time signal is equivalent to the continuous time formulae • They have similar properties to the continuous time Fourier transform for linearity, time shifts, differencing and accumulation • The main result is that like continuous time signals and systems, convolution in the time domain is replaced by multiplication in the frequency domain. Y(ejw) = X(ejw)H(ejw)

11. Example: Unit Pulse and Unit Step • Unit Pulse: • The spectrum is a constant (and periodic over the range [-,]. • Shifted Unit Pulse: • Time delay produces a phase shift linearly proportional to . Note that these functions are also periodic over the range [-,]. • Unit Step: • Since this is not a time-limited function, it hasno DTFT in the ordinary sense. However, it can be shown that the inverse of this function isa unit step:

12. Example: Exponential Decay • Consider an exponentially decaying signal:

13. The Spectrum of an Exponentially Decaying Signal Lowpass Filter: Highpass Filter:

14. Finite Impulse Response Lowpass Filter • The frequency response ofa time-limited pulse is alowpass filter. • We refer to this type offilter as a finite impulse response (FIR) filter. • In the CT case, we obtaineda sinc function (sin(x)/x) for the frequency response. This is close to a sinc function, and is periodic with period 2.

15. Example: Ideal Lowpass Filter (Inverse) The Ideal Lowpass Filter Is Noncausal!

16. Properties of the DTFT • Periodicity: • Linearity: • Time Shifting: • Frequency Shifting: • Example: • Note the roleperiodicityplays in theresult.

17. Properties of the DTFT (Cont.) • Time Reversal: • Conjugate Symmetry: • Also: • Differentiation inFrequency: • Parseval’s Relation: • Convolution:

18. Properties of the DTFT (Cont.) • Time-Scaling:

19. Example: Convolution For A Sinewave

20. Summary • Introduced the Discrete-Time Fourier Transform (DTFT) that is the analog of the Continuous-Time Fourier Transform. • Worked several examples. • Discussed properties:

21. Fast Fourier Transform (FFT)Algorithms

22. Discrete Fourier Transform (DFT) • The DFT provides uniformly spaced samples of the Discrete-Time Fourier Transform (DTFT) • DFT definition: • Requires N2 complex multiplies and N(N-1) complex additions

23. Faster DFT computation? • Take advantage of the symmetry and periodicity of the complex exponential (let WN=e-j2p/N) • symmetry: • periodicity: • Note that two length N/2 DFTs take less computation than one length N DFT: 2(N/2)2<N2 • Algorithms that exploit computational savings are collectively called Fast Fourier Transforms

24. Decimation-in-Time Algorithm • Consider expressing DFT with even and odd input samples: WN=e-j2p/N

25. DIT Algorithm (cont.) • Result is the sum of two N/2 length DFTs • Then repeat decomposition of N/2 to N/4 DFTs, etc. N/2 DFT x[0,2,4,6] X[0…7] N/2 DFT x[1,3,5,7]

26. Detail of “Butterfly” • Cross feed of G[k] and H[k] in flow diagram is called a “butterfly”, due to shape or simplify: -1

27. 8-point DFT Diagram x[0,4,2,6,1,5,3,7] X[0…7]

28. Computation on DSP • Input and Output data • Real data in X memory • Imaginary data in Y memory • Coefficients (“twiddle” factors) • cos (real) values in X memory • sin (imag) values in Y memory • Inverse computed with exponent sign change and 1/N scaling