**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 • the sampling 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

**H [ ]** Output y(n) Input x(n) Discrete-Time System • Discrete-time system • 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