chapter3 Fourier Series Representation of Periodic Signals.pptVIP

Discrete-time Fourier series pair: ak: the Fourier coefficients or the spectral coefficients of x[n] ak=ak+N : We only use N consecutive values of ak in the synthesis equation Synthesis equation Analysis equation * Example 1: sum of a pair sinusoids -periodic with period N=16 * Example 3.12 * and Suppose to get spectrum of x[n] Note: the envelope shape of ak is as * * Note: The Fourier Series representation of a discrete-time periodic signal is a finite series. There are no mathematical issues of convergence. ak—the spectral coefficients of x[n] is periodic with N. The features of the FS ak of periodic discrete-time signal Discrete Harmonic Periodic * 3.7 Properties of Discrete-Time Fourier Series See page 221 Table 3.2 * CT: DT: h(t) est H(s)est h[n] zn H(z)zn System function System function Recall—— 3.8 Fourier Series and LTI System * Frequency response of an LTI system * CT Frequency Response: h(t) ejwt H(jw)ejwt DT Frequency Response: h[n] ejwn H(ejw)ejwn S=jw z=ejw CT: DT: h(t) est H(s)est h[n] zn H(z)zn Fourier Series and LTI System * Example 3.16 (example 3.2) calculate the Fourier Series coefficients of the output y(t). and where * Solution: According to so * Where, Homework: P253- 3.13 3.15 P262-3.34, 3.35 * 3.9 Filtering * filtering frequency-shaping filters frequency-selective filters 3.9.1 Frequency-shaping filters Example1: Equalizer * Example2: Image Filtering (Edge enhancement) * 3.9.2 Frequency-selective filters Several type of filter : (1) Lowpass filter (2) Highpass filter (3) Bandpass filter * three types of filter ( Continuous time ) LPF HPF BPF * three types of filter ( Discrete time ) LPF HPF BPF * Example: Using LPF and HPF to Separate signals Filter design becomes more challenging if the two signals are close in frequency range. * Extraction of narrowband signals from wideband noise – BPF * Filtering out unwanted narrowband signal, for example, 60-Hz power lines –– Notch Filter Apply different filte