About this Course Subject Digital Signal Processing EE
About this Course Subject: ◦ Digital Signal Processing ◦ EE 541 Textbook ◦ Discrete Time Signal Processing ◦ A. V. Oppenheim and R. W. Schafer, Prentice Hall, 3 rd Edition Reference book ◦ Probability and Random Processes with Applications to Signal Processing ◦ Henry Stark and John W. Woods, Prentice Hall, 3 rd Edition Course website ◦ http: //sist. shanghaitech. edu. cn/faculty/luoxl/class/2014 Fall_DSP/DSPclass. htm ◦ Syllabus, lecture notes, homework, solutions etc.
About this Course Grading details: ◦ ◦ Homework: (Weekly) 20% Midterm: 30% Final: 30% Project: 20% Final Score 20% Homework Midterm 30% Final Project
Matlab q Powerful software you will like for the rest of your time in Shanghai. Tech SIST q Ideal for practicing the concepts learnt in this class and doing the final projects
About the Lecturer Name: Xiliang Luo (罗喜良) Research interests: ◦ Wireless communication ◦ Signal processing ◦ Information theory More information: ◦ http: //sist. shanghaitech. edu. cn/faculty/luoxl/
About TA Name: 裴东 Contact: peidong@shanghaitech. edu. cn Office hour: Friday, 6 -8 pm,
Some survey q background q coolest thing you have ever done q what you want to learn from this course?
Lecture 1: Introduction to DSP XILIANG LUO 2014/9
Signals and Systems q Signal q something conveying information q speech signal q video signal q communication signal q continuous time q discrete time q digital signal : not only time is discrete, but also is the amplitude!
Discrete Time Signals
Speech Signal Question: 1. What is the sampling frequency? 2. Are we losing anything here by sampling?
Some Basic Sequences
Some Basic Sequences Question: 1. Is discrete sinusoidal periodic? 2. What is the period? Question: Cos(pi/4 xn) vs Cos(7 pi/4 xn), which One has faster oscillation?
Some Basic Sequences Question: Cos(pi/4 xn) vs Cos(7 pi/4 xn), which One has faster oscillation?
Discrete-Time Systems
Memoryless System Question: Are the following memoryless? 1. y[n] = x[n-d] 2. y[n] = average{x[n-p], …, x[n+q]}
Linear System Additivity Property Scaling Property Superposition Principle
Time-Invariant System q A. k. a. shift-invariant system: a time shift in the input causes a corresponding time shift in the output: Question: Are the following time-invariant? 1. y[n] = x[n-d] 2. y[n] = x[Mn]
Causality q The output of the system at time n depends only on the input sequence at time values before or at time n; Is the following system causal? y[n] = x[n+1] – x[n]
Stability: BIBO Stable q A system is stable in the Bounded-Input, Bounded-Output (BIBO) sense if and only if every bounded input sequence produces a bounded output sequence. q A sequence is bounded if there exists a fixed positive finite value B such that:
LTI Systems q LTI : both Linear and Time-Invariant systems q convenient representation: completely characterized by its impulse response q significant signal-processing applications q Impulse response q LTI System
LTI System q LTI system is completely characterized by its impulse response as follows: convolution sum
Properties of LTI Systems q Commutative: q Distributive: q Associative:
Properties of LTI Systems Equivalent systems:
Properties of LTI Systems Equivalent systems:
Stability of LTI System q LTI systems are stable if and only if the impulse response is absolutely summable: q sufficient condition q need to verify bounded input will have also bounded output under this condition q necessary condition q need to verify: stable system the impulse response is absolutely summable q equivalently: if the impulse response is not absolutely summable, we can prove the system is not stable!
Stability of LTI System q Prove: if the impulse response is not absolutely summable, we can define the following sequence: q x[n] is bounded clearly q when x[n] is the input to the system, the output can be found to be the following and not bounded:
Some Convolution Examples Matlab cmd: conv() what is the resulting shape?
Some Convolution Examples what is the resulting shape?
Some Convolution Examples what is the freq here?
Frequency Domain Representation q Eigenfunction for LTI Systems q complex exponential functions are the eigenfunction of all LTI systems
Frequency Response of LTE Systems q For an LTI system with impulse response h[n], the frequency response is defined as: q In terms of magnitude and phase: phase response magnitude response
Frequency Response of Ideal Delay
Frequency Response for a Real IR q For real impulse response, we can have: why? q Response to a sinusoidal of an LTI with real impulse response
Frequency Response Property
Frequency Response of Typical Filters low pass band-stop high pass band-pass
Representation of Sequences by FT q Many sequences can be represented by a Fourier integral as follows: Synthesis: Inverse Fourier Transform Prove it! Analysis: Discrete-Time Fourier Transform q x[n] can be represented as a superposition of infinitesimally small complex exponentials q Fourier transform is to determine how much of each frequency component is used to synthesize the sequence
Convergence of Fourier Transform q A sufficient condition: absolutely summable q it can be shown the DTFT of absolutely summable sequence converge uniformly to a continuous function
Square Summable q A sequence is square summable if: q For square summable sequence, we have mean-square convergence:
Ideal Lowpass Filter
DTFT of Complex Exponential Sequence q Let a Fourier Transform function be: q Now, let’s find the synthesized sequence with the above Fourier Transform:
Symmetry Properties of DTFT q Conjugate Symmetric Sequence Real even sequence q Conjugate Anti-Symmetric Sequence Real odd sequence q Any sequence can be expressed as the sum of a CSS and a CASS as How?
Symmetry Properties of DTFT q DTFT of a conjugate symmetric sequence is conjugate symmetric q DTFT of a conjugate anti-symmetric sequence is conjugate antisymmetric q Any real sequence’s DTFT is conjugate symmetric
Fourier Transform Theorems q Time shifting and frequency shifting theorem Prove it!
Fourier Transform Theorems q Time Reversal Theorem Prove it!
Fourier Transform Theorems q Differentiation in Frequency Theorem Prove it!
Fourier Transform Theorems q Parseval’s Theorem: time-domain energy = freq-domain energy HW Problem 2. 84: Prove a more general format
Fourier Transform Theorems q Convolution Theorem Prove it!
Fourier Transform Theorems q Windowing Theorem Prove it!
Discrete-Time Random Signals q Wide-sense stationary random process (assuming real) autocorrelation function q Consider an LTE system, let x[n] be the input, which is WSS, the output is denoted as y[n], we can show y[n] is WSS also
Discrete-Time Random Signals q WSS in, WSS out
Discrete-Time Random Signals q WSS in, WSS out
Discrete-Time Random Signals q WSS in, WSS out
Power Spectrum Density band-pass
White Noise q Very widely utilized concept in communication and signal processing q A white noise is a signal for which: q From its PSD, we can see the white noise has equal power distribution over all frequency components q Often we will encounter the term: AWGN, which stands for: additive white Gaussian noise q the underlying random noise is Gaussian distributed
Review q LTI system q Frequency Response q Impulse Response q Causality q Stability q Discrete-Time Fourier Transform q WSS q PSD
Homework Problems 2. 11 Given LTI frequency response, find the output when input a sinusoidal sequence … 2. 17 Find DTFT of a windowed sequence … 2. 22 Period of output given periodic input … 2. 40 Determine the periodicity of signals … 2. 45 Cascade of LTE systems … 2. 51 Check whether system is linear, time-invariant … 2. 63 Find alternative system … 2. 84 General format of Parseval’s theorem … Try to use Matlab to plot the sequences and results when required
Next Week Ø Z – Transform Ø Please read the textbook Chapter 3 in advance!
- Slides: 57