About this Course Subject: Textbook Reference book Course website Digital Signal Processing EE 541 Textbook Discrete Time Signal Processing A. V. Oppenheim and R. W. Schafer, Prentice Hall, 3rd Edition Reference book Probability and Random Processes with Applications to Signal Processing Henry Stark and John W. Woods, Prentice Hall, 3rd Edition Course website http://sist.shanghaitech.edu.cn/faculty/luoxl/class/2014Fall_DSP/DSPclass.htm Syllabus, lecture notes, homework, solutions etc.

About this Course Grading details: Homework: (Weekly) 20% Midterm: 30% Final: 30% Project: 20%

Matlab Powerful software you will like for the rest of your time in ShanghaiTech SIST 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, 6-8pm,

Some survey background coolest thing you have ever done what you want to learn from this course?

Lecture 1: Introduction to DSP Xiliang Luo 2014/9

Signals and Systems Signal something conveying information speech signal video signal communication signal continuous time discrete time digital signal : not only time is discrete, but also is the amplitude!

Discrete Time Signals Mathematically, discrete-time signals can be expressed as a sequence of numbers 𝑥 𝑛 ,𝑛∈𝑍 In practice, we obtain a discrete-time signal by sampling a continuous- time signal as: 𝑥 𝑛 = 𝑥 𝑎 (𝑛𝑇) where T is the sampling period and the sampling frequency is defined as 1/T

Speech Signal Question: 1. What is the sampling frequency? 2. Are we losing anything here by sampling?

Some Basic Sequences Unit Sample Sequence 𝛿 𝑛 = 0, 𝑛≠0 1, 𝑛=0 𝛿 𝑛 = 0, 𝑛≠0 1, 𝑛=0 Unit Step Sequence 𝛿 𝑛 = 0, 𝑛<0 1, 𝑛≥0

Some Basic Sequences Sinusoidal Sequence x 𝑛 =𝐴 cos ( 𝜔 0 𝑛+𝜙) Question: Is discrete sinusoidal periodic? What is the period? Question: Cos(pi/4xn) vs Cos(7pi/4xn), which One has faster oscillation?

Some Basic Sequences Sinusoidal Sequence x 𝑛 =𝐴 cos ( 𝜔 0 𝑛+𝜙) Question: Cos(pi/4xn) vs Cos(7pi/4xn), which One has faster oscillation?

Discrete-Time Systems a transformation or operator mapping discrete time input to discrete time output 𝑦 𝑛 =𝑇{𝑥[𝑛]} Example: ideal delay system y[n] = x[n-d] Example: moving average y[n] = average{x[n-p],….,x[n+q]}

Memoryless System Definition: output at time n depends only on the input at the sample time n 𝑦 𝑛 =𝑥 𝑛 2 Question: Are the following memoryless? y[n] = x[n-d] y[n] = average{x[n-p], …, x[n+q]}

Linear System Definition: systems satisfying the principle of superposition 𝑇 𝑥 1 𝑛 + 𝑥 2 𝑛 =𝑇 𝑥 1 [𝑛] +𝑇{ 𝑥 2 [𝑛]} 𝑇 𝑎𝑥[𝑛] =𝑎𝑇 𝑥[𝑛] 𝑇 𝑎 𝑥 1 𝑛 + 𝑏𝑥 2 𝑛 =𝑎𝑇 𝑥 1 [𝑛] +𝑏𝑇{ 𝑥 2 [𝑛]} Additivity Property Scaling Property Superposition Principle

Time-Invariant System A.k.a. shift-invariant system: a time shift in the input causes a corresponding time shift in the output: 𝑇 𝑥[𝑛] =𝑦[𝑛] 𝑇 𝑥[𝑛−𝑑] =𝑦[𝑛−𝑑] Question: Are the following time-invariant? y[n] = x[n-d] y[n] = x[Mn]

Causality 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 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. A sequence is bounded if there exists a fixed positive finite value B such that: 𝑥 𝑛 ≤𝐵<∞

LTI Systems LTI : both Linear and Time-Invariant systems convenient representation: completely characterized by its impulse response significant signal-processing applications Impulse response LTI System ℎ 𝑛 =𝑇{𝛿[𝑛]} 𝑥 𝑛 = 𝑘 𝑥 𝑘 𝛿[𝑛−𝑘] 𝑦 𝑛 =𝑇 𝑘 𝑥 𝑘 𝛿[𝑛−𝑘] = 𝑘 𝑥 𝑘 𝑇{𝛿 𝑛−𝑘 ] = 𝑘 𝑥 𝑘 ℎ[𝑛−𝑘]

LTI System LTI system is completely characterized by its impulse response as follows: ℎ 𝑛 =𝑇{𝛿[𝑛]} 𝑦 𝑛 = 𝑘 𝑥 𝑘 ℎ 𝑛−𝑘 convolution sum ≜𝑥 𝑛 ∗ℎ[𝑛]

Properties of LTI Systems Commutative: Distributive: Associative: 𝑥 𝑛 ∗ℎ 𝑛 =ℎ 𝑛 ∗𝑥[𝑛] 𝑥 𝑛 ∗ ℎ 1 𝑛 + ℎ 2 𝑛 =𝑥 𝑛 ∗ ℎ 1 𝑛 +𝑥 𝑛 ∗ ℎ 2 [𝑛] (𝑥 𝑛 ∗ ℎ 1 𝑛 )∗ ℎ 2 𝑛 =𝑥 𝑛 ∗( ℎ 1 𝑛 ∗ ℎ 2 [𝑛])

Properties of LTI Systems Equivalent systems:

Properties of LTI Systems Equivalent systems:

Stability of LTI System LTI systems are stable if and only if the impulse response is absolutely summable: sufficient condition need to verify bounded input will have also bounded output under this condition necessary condition need to verify: stable system  the impulse response is absolutely summable equivalently: if the impulse response is not absolutely summable, we can prove the system is not stable! 𝑘=−∞ +∞ |ℎ[𝑘]| <∞

Stability of LTI System Prove: if the impulse response is not absolutely summable, we can define the following sequence: x[n] is bounded clearly when x[n] is the input to the system, the output can be found to be the following and not bounded: 𝑥 𝑛 = ℎ ∗ [−𝑛] |ℎ[−𝑛]| , ℎ −𝑛 ≠0 0, ℎ −𝑛 =0 𝑦 0 = 𝑥 𝑘 ℎ −𝑘 = ℎ 𝑘 2 |ℎ[𝑘]|

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? sin⁡( 𝑛𝜋 8 )

Frequency Domain Representation Eigenfunction for LTI Systems complex exponential functions are the eigenfunction of all LTI systems 𝑦 𝑛 = 𝑒 𝑗𝜔𝑛 ∗ℎ 𝑛 = 𝑘 ℎ 𝑘 𝑒 𝑗𝜔 𝑛−𝑘 = 𝑒 𝑗𝜔𝑛 × 𝑘 ℎ 𝑘 𝑒 −𝑗𝜔𝑘 𝐻( 𝑒 𝑗𝜔 )= 𝑘 ℎ 𝑘 𝑒 −𝑗𝜔𝑘 𝑦 𝑛 =𝐻 𝑒 𝑗𝜔 𝑒 𝑗𝜔𝑛

Frequency Response of LTE Systems For an LTI system with impulse response h[n], the frequency response is defined as: In terms of magnitude and phase: 𝐻( 𝑒 𝑗𝜔 )= 𝑘 ℎ 𝑘 𝑒 −𝑗𝜔𝑘 phase response 𝐻( 𝑒 𝑗𝜔 )= 𝐻 𝑒 𝑗𝜔 𝑒 ∠𝐻( 𝑒 𝑗𝜔 ) magnitude response

Frequency Response of Ideal Delay ℎ 𝑛 =𝛿[𝑛− 𝑛 𝑑 ] 𝐻 𝑒 𝑗𝜔 = 𝑛 𝛿 𝑛− 𝑛 𝑑 𝑒 −𝑗𝜔𝑛 = 𝑒 −𝑗𝜔 𝑛 𝑑

Frequency Response for a Real IR For real impulse response, we can have: Response to a sinusoidal of an LTI with real impulse response 𝐻 𝑒 −𝑗𝜔 = 𝐻 ∗ ( 𝑒 𝑗𝜔 ) why? 𝑥 𝑛 = Acos ( 𝜔 0 𝑛+𝜙 ) = 𝐴 2 𝑒 𝑗(𝜙+ 𝜔 0 𝑛) + 𝐴 2 𝑒 −𝑗(𝜙+ 𝜔 0 𝑛) 𝑦 𝑛 = 𝐴 2 𝐻 𝑒 𝑗 𝜔 𝑜 𝑒 𝑗(𝜙+ 𝜔 0 𝑛) + 𝐴 2 𝐻( 𝑒 −𝑗 𝜔 0 ) 𝑒 −𝑗(𝜙+ 𝜔 0 𝑛) = 𝐴 2 |𝐻 𝑒 𝑗 𝜔 𝑜 | 𝑒 𝑗 𝜙+ 𝜔 0 𝑛+∠𝐻 𝑒 𝑗 𝜔 𝑜 + 𝐴 2 |𝐻 𝑒 𝑗 𝜔 0 | 𝑒 −𝑗(𝜙+ 𝜔 0 𝑛+∠𝐻 𝑒 𝑗 𝜔 𝑜 ) = 𝐻 𝑒 𝑗 𝜔 𝑜 𝐴 cos ( 𝜔 0 𝑛+𝜙+∠𝐻( 𝑒 𝑗 𝜔 0 ))

Frequency Response Property Frequency response is periodic with period 2π fundamentally, the following two discrete frequencies are indistinguishable 𝜔, 𝜔+2𝜋 We only need to specify frequency response over an interval of length 2π : [- π, + π]; In discrete time: low frequency means: around 0 high frequency means: around +/- π

Frequency Response of Typical Filters low pass band-stop high pass band-pass

Representation of Sequences by FT Many sequences can be represented by a Fourier integral as follows: x[n] can be represented as a superposition of infinitesimally small complex exponentials Fourier transform is to determine how much of each frequency component is used to synthesize the sequence 𝑥 𝑛 = 1 2𝜋 −𝜋 𝜋 𝑋 𝑒 𝑗𝜔 𝑒 𝑗𝜔𝑛 𝑑𝜔 Synthesis: Inverse Fourier Transform Prove it! 𝑋 𝑒 𝑗𝜔 = 𝑛 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛 Analysis: Discrete-Time Fourier Transform

Convergence of Fourier Transform A sufficient condition: absolutely summable it can be shown the DTFT of absolutely summable sequence converge uniformly to a continuous function

Square Summable A sequence is square summable if: For square summable sequence, we have mean-square convergence: 𝑛=−∞ ∞ 𝑥[𝑛] 2 <∞

Ideal Lowpass Filter

DTFT of Complex Exponential Sequence Let a Fourier Transform function be: Now, let’s find the synthesized sequence with the above Fourier Transform:

Symmetry Properties of DTFT Conjugate Symmetric Sequence Conjugate Anti-Symmetric Sequence Any sequence can be expressed as the sum of a CSS and a CASS as 𝑥 𝑒 𝑛 = 𝑥 𝑒 ∗ [−𝑛] Real  even sequence 𝑥 𝑜 𝑛 = −𝑥 𝑜 ∗ [−𝑛] Real  odd sequence 𝑥 𝑛 =𝑥 𝑒 𝑛 + 𝑥 𝑜 [𝑛] How?

Symmetry Properties of DTFT DTFT of a conjugate symmetric sequence is conjugate symmetric DTFT of a conjugate anti-symmetric sequence is conjugate anti- symmetric Any real sequence’s DTFT is conjugate symmetric

Fourier Transform Theorems Time shifting and frequency shifting theorem Prove it!

Fourier Transform Theorems Time Reversal Theorem Prove it!

Fourier Transform Theorems Differentiation in Frequency Theorem Prove it!

Fourier Transform Theorems Parseval’s Theorem: time-domain energy = freq-domain energy HW Problem 2.84: Prove a more general format

Fourier Transform Theorems Convolution Theorem Prove it!

Fourier Transform Theorems Windowing Theorem Prove it!

Discrete-Time Random Signals Wide-sense stationary random process (assuming real) 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 𝜙 𝑥𝑥 𝑛,𝑚 =𝐸 𝑥 𝑛 𝑥[𝑛+𝑚] = 𝜙 𝑥𝑥 [𝑚] autocorrelation function

Discrete-Time Random Signals WSS in, WSS out

Discrete-Time Random Signals WSS in, WSS out

Discrete-Time Random Signals WSS in, WSS out

Power Spectrum Density band-pass

White Noise Very widely utilized concept in communication and signal processing A white noise is a signal for which: From its PSD, we can see the white noise has equal power distribution over all frequency components Often we will encounter the term: AWGN, which stands for: additive white Gaussian noise the underlying random noise is Gaussian distributed 𝜙 𝑥𝑥 𝑚 = 𝜎 𝑥 2 𝛿[𝑚] Φ 𝑥𝑥 𝑒 𝑗𝜔 = 𝜎 𝑥 2

Review LTI system Frequency Response Impulse Response Causality Stability Discrete-Time Fourier Transform WSS 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 Z – Transform Please read the textbook Chapter 3 in advance!