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

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

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

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

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.

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

Highpass Filters Remember: high frequency highest frequency in DT

Bandpass Filters Demo:Filtering effects on audio signals

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

Highpass CT DT

Bandpass CT lower cut-off upper cut-off DT

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

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

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

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

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

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

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