数字信号处理教学课件：2-1 Time-Domain Representation

1、Discrete-time signals xn-represented as sequences with argument n being an integer in the range - n The nth sample value of sequence xn is also denoted as xnvalues of argument n-defined only for integer and undefined for non-integer values of nDiscrete-time signal represented by xnxn is a real seque

2、nce, if the n-th sample xn is real for all values of nOtherwise, xn is a complex sequence2 Discrete-Time Signals In the Time-Domain2.1 Time-Domain Representation1Discrete-time signal may also be written as a sequence of numbers inside braces: xn=,-0.2, 2.2, 1.1, 0.2, -3.7, 2.9, The arrow is placed u

3、nder the sample at time index n = 0In the above, x-1= -0.2, x0=2.2, x1=1.1, etc. 2 Discrete-Time Signals In the Time-Domain2Graphical representation of a discrete-time signal with real-valued samples is as shown below:2 Discrete-Time Signals In the Time-Domain3In some applications, a discrete-time s

4、equence xn may be generated by periodically sampling a continuous-time signal xa(t) at uniform intervals of time2 Discrete-Time Signals In the Time-Domain4Here, n-th sample is given by xn=xa(t) |t=nT=xa(nT), n=,-2,-1,0,1, The spacing T between two consecutive samples is called the sampling interval

5、or sampling periodReciprocal of sampling interval T, denoted as FT , is called the sampling frequency: FT=1/T2 Discrete-Time Signals In the Time-Domain5Unit of sampling frequency is cycles per second, or hertz (Hz), if T is in secondsWhether or not the sequence xn has been obtained by sampling, the

6、quantity xn is called the n-th sample of the sequencexn is a real sequence, if the n-th sample xn is real for all values of nOtherwise, xn is a complex sequence2 Discrete-Time Signals In the Time-Domain6A complex sequence xn can be written as xn=xren+jximn (2.4) where xren and ximn are the real and

7、imaginary parts of xnThe complex conjugate sequence of xn is given by x*n=xren - jxim n Often the braces are ignored to denote a sequence if there is no ambiguity2 Discrete-Time Signals In the Time-Domain7Example - xn=cos0.25n is a real sequence yn=ej0.3n is a complex sequenceWe can write yn=cos0.3n

8、 + jsin0.3n = cos0.3n + jsin0.3n where yren=cos0.3n yimn=sin0.3n2 Discrete-Time Signals In the Time-Domain8Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued- Digital signals in which samples are discrete-valuedSignals in a practical digital signal proc

9、essing system are digital signals obtained by quantizing the sample values either by rounding or truncation2 Discrete-Time Signals In the Time-Domain9ExampleAmplitudeDigital signalAmplitudeBoxcar signalTime,tTime,t2 Discrete-Time Signals In the Time-Domain10A discrete-time signal may be a finite-len

10、gth or an infinite-length sequenceFinite-length (also called finite-duration or finite-extent) sequence is defined only for a finite time interval:N1 n N2where - N1 and N2 with N1 N2Length or duration of the above finite-length sequence is N= N2 - N1+ 1Zero-paddingappend with zero-valued samples.2.1

11、.1 Length of a discrete-time signal Vector form xn=x0 x1 xN-1t2 Discrete-Time Signals In the Time-Domain11A right-sided sequence xn has zero-valued samples for n N2If N2 0, a left-sided sequence is called a anticausal sequence.A left-sided sequence2 Discrete-Time Signals In the Time-Domain132.1.2 St

12、rength of a Discrete-Time SignalIn practice, the value of p used is typically 1 or 2 or .The strength of a DT Signal usually is given by its norm. of xn is defied by 2 Discrete-Time Signals In the Time-Domainwith142.1.2 Strength of a Discrete-Time Signal2 Discrete-Time Signals In the Time-DomainIllu

13、strations ofunit circlesin differentp-norms15For a length-N sequence, is the root-mean-squared (rms) value of xn, and is the mean absolute value of xn.It can be shown thatnorm(x,1)norm(x,2)norm(x,inf)MATLAB functions2 Discrete-Time Signals In the Time-Domain16(2.11)For example, a length-N sequence y

14、n, another xn,One application of the norm is in estimating the error in approximation of a DT signal by another DT signal in some sense.0 n N-1mean-squared error(MSE)relative error(2.12)2 Discrete-Time Signals In the Time-Domain172.2 The Basic Operations On Sequences 2.2.1 Elementary Operations Modu

15、lator (product or windowing) nxny1nynxnw=modulatornx2nAxnw=multipler18pick-off node2 Discrete-Time Signals and Systemsaddition(subtraction) nxny3nynxnw+=adder1-Znx14-=nxnwunit delayZnx15+=nxnwunit advance time-reversal (folding operation)x-n19Example 2.1 Ensemble average (Application of addition operation)2 Discrete-Time Signals and Systemsoriginal uncorrupted data Noise20Ensemble average2 Di