1. Signals and Systems Fall 2003 Lecture #7 25 September 2003 1.Fourier Series and LTI Systems 2.Frequency Response and Filtering 3.Examples and Demos

2. The Eigenfunction Property of Complex Exponentials CT: DT: "System Function" CT DT "System Function"

3. Fourier Series: Periodic Signals and LTI Systems So or powers of signals get modified through filter/system Includes both amplitude & phase

4. The Frequency Response of an LTI System CT Frequency response: DT Frequency response:

5. Frequency Shaping and Filtering By choice of H(j ω ) (or H(e j ω )) as a function of ω, we can shape the frequency composition of the output - Preferential amplification - Selective filtering of some frequencies Example #1: Audio System Adjustable Equalizer Speaker Filter Bass, Mid-range, Treble controls For audio signals, the amplitude is much more important than the phase.

6. Example #2:Frequency Selective Filters Filter out signals outside of the frequency range of interest Lowpass Filters: Only show amplitude here. Note for DT: Stopband PassbandStopband

7. Highpass Filters Remember: high frequency highest frequency in DT

8. Bandpass Filters Demo:Filtering effects on audio signals

9. Idealized Filters CT DT Note: |H| = 1 and H = 0 for the ideal filters in the passbands, no need for the phase plot. ω c cutoff frequency Stopband PassbandStopband

10. Highpass CT DT

11. Bandpass CT lower cut-off upper cut-off DT

12. Example #3:DT Averager/Smoother FIR (Finite Impulse Response) filters LPF

13. Example #4:Nonrecursive DT (FIR) filters Rolls off at lower ω as M+N+1 increases

14. Example #5:Simple DT Edge Detector DT 2-point differentiator Passes high-frequency components

15. Demo:DT filters, LP, HP, and BP applied to DJ Industrial average Original signFiltered sign Dollars Year

16. Example #6: Edge enhancement using DT differentiator Courtesy of Jason Oppenheim. Used with permission.

17. Example #7:A Filter Bank HPF BPF #1 BPF #M LPF

18. Demo:Apply different filters to two-dimensional image signals. Face of a monkey. Note: To really understand these examples, we need to understand frequency contents of aperiodic signals the Fourier Transform Image removed do to copyright considerations