ECE4270 Fundamentals of DSP Lecture 7-B Discrete-Time Fourier Transform Theorems

1. ECE4270Fundamentals of DSPLecture 7-BDiscrete-Time Fourier Transform Theorems School of Electrical and Computer Engineering Center for Signal and Image Processing Georgia Institute of Technology

2. Overview of Lecture 7-B • Announcement • Definition of DTFT • Symmetry properties of the DTFT • Importance of theorems • The time-domain convolution theorem • Examples of usage of the DTFT

3. Announcement • Quiz # 1 • Monday Feb. 20 (at the class hour). • Open book and notes (closed HWKs). • Coverage: Chapters 2 and 3.

4. Direct Transform Inverse Transform Discrete-Time Fourier Transform • Definition: • The frequency response function is now seen to be just the DTFT of the impulse response.

5. We worked this one out. And this one too.

6. DTFT of a Constant Signal • Assume a “periodic impulse train” for the DTFT • Find the corresponding sequence x[n]

7. Very Useful DTFT Pairs

9. DTFT of Sinusoids • Recall that • Also note that • Therefore

10. LTI System Convolution Theorem

11. Example 1 • Find the output when the input is

12. Example 2 • Find the output when the input is

13. Frequency Response of a DE

14. Example 3 • Suppose that the difference equation is • The frequency response is • This system is implemented in MATLAB by • >> y=filter([1,1],[1,-1,.9],x) • We can compute its frequency response by • >> omega=(0:500)*pi/500; • >> H=freqz ([1,1],[1,-1,.9],omega);

15. Example (revisit) • Consider the real exponential signal • Its DTFT is obtained as follows:

16. Real and Imaginary Parts

17. Magnitude and Angle Form

18. Magnitude and Angle Plots