CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain YANG Jian jianyang@ynu.edu.cn School of Information Science and Technology Yunnan University

Outline Discrete-Time Signals Typical Sequences and Sequence Representation The Sampling Process Discrete-Time Systems Time-Domain Characterization of LTI Discrete-Time Systems Finite-Dimensional LTI Discrete-Time Systems Correlation of Signals Summary 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time Signals Basic signals Unit sample or unit impulse sequence Unit step sequence Exponential sequence Signal classification Continuous-time / discrete-time signals Deterministic / random signals Energy signals signals with finite energy Power signals signals with finite power Energy signals have zero power, and power signals have infinite energy 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time Signals Time-Domain Representation Sequence of numbers: — sequence — samples — sample value or nth samples, a real or complex value Figure of sequence: is defined only for integer value of 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time Signals Operation on sequences Basic operation Adder / Subtraction: Scalar multiplication ( gain / attenuation ): Delay / Advance: Combination of Basic Operations Multiplier: Linear combination: 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time Signals Operation on sequences Sampling Rate Alteration ( special operations of for discrete-time signals ) Up-sampling: Down-sampling: 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time Signals Classification of Sequences The number of sequences: finite / infinite Finite-length sequences: Symmetry conjugate-symmetric ( even ): conjugate-antisymmetric ( odd ): Periodity: periodic / aperiodic Periodic sequence: 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time Signals Classification of Sequences Energy and Power Signals 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time Signals Classification of Sequences Other types of Classification Bounded: Absolutely summable: Square-summable: 云南大学滇池学院课程：现代信号处理数字信号处理

Typical Sequences and Sequence Representation Some Basic Sequences Unite sample sequence: An arbitrary sequence can be represented by unite sample sequence in time-domain Unite step sequence: 云南大学滇池学院课程：现代信号处理数字信号处理

Typical Sequences and Sequence Representation Sinusoidal and Exponential Sequences The real sinusoidal sequence: The exponential sequence: The sinusoidal sequence are periodic of period N as long as is an integer multiple of . The smallest possible N is the fundamental period of the sequence. 云南大学滇池学院课程：现代信号处理数字信号处理

Typical Sequences and Sequence Representation Some Typical Sequences Regular window sequence: Real exponential sequence: Representation of an Arbitrary Sequence An arbitrary sequence can be represented as a weight sum of basic sequence and its delayed version. 云南大学滇池学院课程：现代信号处理数字信号处理

The Sampling Process Uniform sampling: Often the discrete-time sequence is developed by uniformly sampling a continuous-time signal : the sampling frequency angular frequency 云南大学滇池学院课程：现代信号处理数字信号处理

The Sampling Process Aliasing: When , a continuous-time sinusoidal signal of higher frequency would acquire the identity of a sinusoidal sequence of lower frequency after sampling. e.g. 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time System Discrete-time system H [ ] Output y(n) Input x(n) Simple Discrete-Time Systems The accumulator The M-point moving-average filter The factor-of-L interpolator 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time System Classification of Discrete-Time System Linear system: Shift-Invariant System: LTI System: The linear time-invariable discrete-time system satisfies both the linear and the time-invariable properties. 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time System Classification of Discrete-Time System Causal System: In a causal discrete-time system, the th output sample depends only on input samples for and does notdepend on input samples for . 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time System Classification of Discrete-Time System Stable System: Definition of bounded-input, bounded-output ( BIBO ) stable. Passive and Lossless Systems The passivity: The losslessness: the same energy 云南大学滇池学院课程：现代信号处理数字信号处理

Discrete-Time System Impulse and Step Responses Input sequence → output sequence Impulse response : Step response : 云南大学滇池学院课程：现代信号处理数字信号处理

Time-Domain Characterization of LTI Discrete-Time Systems Input-Output Relationship The response y(n) of the LTI discrete-time system to x(n) will be given by the convolution sum: The operation Step 1, time-reverse: Step 2, shift n sampling period: Step 3, product: Step 4, summing all samples: 云南大学滇池学院课程：现代信号处理数字信号处理

Time-Domain Characterization of LTI Discrete-Time Systems Some useful properties of the convolution operation Commutative: Associative for stable and single-sided sequences: Distributive: 云南大学滇池学院课程：现代信号处理数字信号处理

Time-Domain Characterization of LTI Discrete-Time Systems Simple Interconnection Schemes Cascade Connection: Parallel Connection: Inverse System: 云南大学滇池学院课程：现代信号处理数字信号处理

Time-Domain Characterization of LTI Discrete-Time Systems Stability Condition in Terms of the Impulse Response An LTI digital filter is BIBO stable if only if its impulse response sequence is absolutely summable, i.e.: Causality Condition in Terms of the Impulse Response An LTI discrete-time system is causal if and only if its impulse response is a causal sequence satisfying the condition: 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems The difference equation: An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form: The order of the system is given by max( N, M ) 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Total Solution Calculation The complementary solution The homogeneous difference equation: The characteristic equations: 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Total Solution Calculation The particular solution is of the same form as specified input . The total solution: 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Zero-Input Response and Zero-State Response zero-input response = complementary solution with initials; zero-state response = the convolution sum of x(n) and h(n). 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Impulse Response Calculation 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Impulse Response Calculation The solutions 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Location of Roots of Characteristic Equation for BIBO Stability A casual LTI system characteristic of a linear constant coefficient difference equation is BIBO stable, if the magnitude of each of the roots its characteristic equation is less than 1. The necessary and sufficient condition: 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Classification of LTI System Based on impulse response length Finite impulse response ( FIR ): Infinite impulse response ( IIR ): 云南大学滇池学院课程：现代信号处理数字信号处理

Finite-Dimensional LTI Discrete-Time Systems Classification of LTI System Based on the output calculation process Non-recursive system: If the output sample can be calculated sequentially, knowing only the present and pass input samples. Recursive system: If the computation of the output involves past output samples. Remarks: FIR — Non-recursive IIR — Recursive 云南大学滇池学院课程：现代信号处理数字信号处理

Correlation of Signals Definitions A measure of similarity between a pair of energy signals, x(n) and y(n), is given by the cross-correlation sequence defined by: The autocorrelation sequence of x(n) is given by: 云南大学滇池学院课程：现代信号处理数字信号处理

Correlation of Signals Properties of Autocorrelation and Cross-correlation Sequences Set and as energies of the sequences x(n) and y(n) , then we can get or equivalently If y(n) = x(n), then The sample value of the autocorrelation sequence has its max value at zero lag ( l = 0 ). 云南大学滇池学院课程：现代信号处理数字信号处理

Correlation of Signals Properties of Autocorrelation and Cross-correlation Sequences If , where N is integer and b>0 is an arbitrary number. In this case , so 云南大学滇池学院课程：现代信号处理数字信号处理

Correlation of Signals Normalized Forms of Correlation: Correlation Computation for Power and Periodic Signals Power signals: Periodic signals: 云南大学滇池学院课程：现代信号处理数字信号处理

Summary The LTI system has numerous applications in practice. The LTI system can be described by an input-output relation composed of a linear constant coefficient difference equation. The LTI discrete-time system is usually classified in terms of the length of its impulse response. 云南大学滇池学院课程：现代信号处理数字信号处理