Signals and Systems Lecture 9 Fourier Series and LTI Systems Frequency Response and Filtering

Chapter 3 Fourier Series §3.8 Fourier Series and LTI Systems Eigenfunctions of LTI System System Function System Function

Chapter 3 Fourier Series §3.8 Fourier Series and LTI Systems Frequency Response of an LTI System Frequency Response of CT LTI Systems Frequency Response of DT LTI Systems

Chapter 3 Fourier Series §3.8 Fourier Series and LTI Systems Continuous-time LTI System Linear Combinations of Eigenfunctions Periodic Signal Frequency Response of LTI System including both amplitude & phase must be well defined and finite.

Example:3.16 Chapter 3 Fourier Series Input signal: Impulse response: Find out bk of output signal Frequency response:

Chapter 3 Fourier Series Discrete-time LTI System Linear Combinations of Eigenfunctions Periodic Signal Frequency Response of LTI System including both amplitude & phase must be well defined and finite.

Example:3.17 Chapter 3 Fourier Series Input signal: Impulse response: Find out the output signal Frequency response: If N=4, we get then and

Chapter 3 Fourier Series Example Consider an LTI system with input the unit impulse response , determine the Fourier Series Representation of output 0 k is even k is odd

Frequency-Shaping Filter Frequency-Selective Filter Chapter 3 Fourier Series §3.9 Filtering（滤波） Filter Frequency-Shaping Filter 频率成形滤波器 Frequency-Selective Filter 频率选择性滤波器

Chapter 3 Fourier Series §3.9.1 Frequency-Shaping Filter Change the shape of the spectrum 1. Equalizer（均衡器）

Chapter 3 Fourier Series 2. Differentiator （微分器） D

Chapter 3 Fourier Series Figure 3.24

Chapter 3 Fourier Series 3. Discrete-time system

Chapter 3 Fourier Series §3.9.2 Frequency-Selective Filter Filter out signals outside of the frequency range of interest Lowpass Filters: Only show amplitude here. Note for DT:

Chapter 3 Fourier Series §3.9.2 Frequency-Selective Filter Highpass Filters Remember: — = highest frequency in DT

Chapter 3 Fourier Series §3.9.2 Frequency-Selective Filter Bandpass Filters

Chapter 3 Fourier Series §3.9.2 Frequency-Selective Filter 1. Ideal Lowpass Filter (LPF) Continuous-time system Stopband Passband Discrete-time system Stopband Passband

Chapter 3 Fourier Series 2. Ideal Highpass Filter (HPF) Continuous-time system Passband Stopband Discrete-time system

Chapter 3 Fourier Series 3. Ideal Bandpass Filter (BPF) Continuous-time system Discrete-time system

Chapter 3 Fourier Series Example Consider an LTI system with input the frequency response of this system is as shown in Figure 1 , determine the output of system Figure 1 (a) Figure 1 (b)

Chapter 3 Fourier Series §3.10 Examples of continuous-time filters described by differential equations §3.10.1 A simple RC Lowpass Filter

Chapter 3 Fourier Series §3.10.2 A simple RC Highpass Filter

Chapter 3 Fourier Series §3.11 Examples of discrete-time filters described by difference equations §3.11.1 First-Order Recursive DT Filter

Chapter 3 Fourier Series §3.11.2 Nonrecursive DT Filters Example : DT Averager/Smoother

Chapter 3 Fourier Series §3.11.2 Nonrecursive DT Filters Example : DT Averager/Smoother Rolls off at nω lower as M+N+1 increases

Example : Simple DT “Edge” Detector Chapter 3 Fourier Series Example : Simple DT “Edge” Detector DT 2-points “differentiator” Amplifies high-frequency components

Summary Fourier Series and LTI Systems Frequency Response and Filtering

Readlist Signals and Systems: Chapter1~Chapter3

Chapter 3 Fourier Series Homework: 3.15 3.35

Chapter 3 Fourier Series Homework: 3.1 3.13 3.15 3.34 3.35

Chapter 3 Fourier Series Eigenfunction Eigenvalue

Synthesis equation 综合公式 Analysis equation 分析公式 Chapter 3 Fourier Series 1. Periodic square wave 2. Periodic Impulse Trains

Chapter 3 Fourier Series

Chapter 3 Fourier Series real odd Purely imaginary odd real even