Presentation on theme: "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."— Presentation transcript:
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
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