Chapter 2 Discrete-Time Signals and Systems
Chapter 2 Discrete-Time Signals and Systems
2.1 Discrete-Time Signals: Time-Domain Representation Signals represented as sequences of numbers,called samples In some applications,a discrete-time sequence [n} may be generated by periodically sampling a continuous-time signal x(t)at uniform intervals of time xa(-5T 37 -5T -3T-T 0 T xa(3T)
§2.1 Discrete-Time Signals: Time-Domain Representation Signals represented as sequences of numbers, called samples In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal xa(t) at uniform intervals of time
2.1 Discrete-Time Signals: Time-Domain Representation Here,n-th sample is given by x[n=xa()l=nr=xa(nT),n=.,-2,-1,0,1,. The spacing T between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval T,denoted as Fr,is called the sampling frequency: F=1/T
§2.1 Discrete-Time Signals: Time-Domain Representation • Here, n-th sample is given by x[n]=xa(t)| t=nT=xa(nT), n=…,-2,-1,0,1,… • The spacing T between two consecutive samples is called the sampling interval or sampling period • Reciprocal of sampling interval T, denoted as FT , is called the sampling frequency: FT=1/T
2.1 Discrete-Time Signals: Time-Domain Representation Two types of discrete-time signals: Sampled-data signals in which samples are continuous-valued Digital signals in which samples are discrete-valued Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation
§2.1 Discrete-Time Signals: Time-Domain Representation • Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued - Digital signals in which samples are discrete-valued • Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation
2.1 Discrete-Time Signals: Time-Domain Representation A right-sided sequence xn]has zero- valued samples for nN 00 A right-sided sequence .If N >0,a right-sided sequence is called a causal sequence
§2.1 Discrete-Time Signals: Time-Domain Representation • A right-sided sequence x[n] has zerovalued samples for n < N1 n N1 A right-sided sequence •If N1≥ 0, a right-sided sequence is called a causal sequence
2.2 Operations on Sequences A single-input,single-output discrete- time system operates on a sequence, called the input sequence,according some prescribed rules and develops another sequence,called the output sequence,with more desirable properties x Discrete-time system y[n] Input sequence Output sequence
§2.2 Operations on Sequences • A single-input, single-output discretetime system operates on a sequence, called the input sequence, according some prescribed rules and develops another sequence, called the output sequence, with more desirable properties x[n] y[n] Input sequence Output sequence Discrete-time system
S 2.2.1 Basic Operations Product (modulation)operation: Modulator x 叫一一 y叫 y[n]=x[n].w[n] w[n] Addition operation: Adder x[n] 一9一 y[n]y[n]=x[n]+w[n] w[n] Multiplication operation Multiplier x[n]-一yy回=A.xnl
§2.2.1 Basic Operations • Product (modulation) operation: Modulator x[n] × y[n] w[n] y[n]=x[n].w[n] • Addition operation: x[n] y[n] w[n] Adder + y[n]=x[n]+w[n] • Multiplication operation A Multiplier x[n] y[n] y[n]=A.x[n]
S 2.2.1 Basic Operations Time-shifting operation,where N is an integer If N>0,it is delaying operation -Unit delay xin] y[n]=x[n-1] If N<0,it is an advance operation -Unit advance x[n]- 一y[n y[]=x[n+1]
§2.2.1 Basic Operations • Time-shifting operation, where N is an integer • If N > 0, it is delaying operation −1 x[n] z y[n] –Unit delay y[n]=x[n-1] x[n] z y[n] -Unit advance y[n]=x[n+1] If N < 0, it is an advance operation
S 2.2.1 Basic Operations Time-reversal (folding)operation: y[n]=x[-n] Branching operation:Used to provide multiple copies of a sequence x x叫 x n]
§2.2.1 Basic Operations • Time-reversal (folding) operation: y[n]=x[-n] • Branching operation: Used to provide multiple copies of a sequence x[n] x[n] x[n]
S 2.2.1 Basic Operations Example- x[n-] 1 x[n-2] -1 x[n-3] x[n] y(n] yln=o1xn+02x[n-1]+3ln-2]+4xn-3]
§2.2.1 Basic Operations • Example - y[n]=α1x[n]+ α2x[n-1]+ α3[n-2]+ α4x[n-3]
