1、1Chapter 6Time and Frequency Characterizationof Signals and Systems2 th txjHjX tyjY Chapter 6 Time and Frequency Characterization thtxtyTime-Domain:jHjXjYFrequency-Domain:6.1 The Magnitude-Phase Representation （幅度幅度-相位）相位）of the Fourier Transform jXjejXjX dejXtxtj 21 jXMagnitude 2 jXEnergy-Density3
2、Chapter 6 Time and Frequency Characterization6.2 The Magnitude-Phase Representation of the Frequency Response of LTI Systems jHjXjY jHjXjY jHjXjY jHGain jHPhase Shift4 Chapter 6 Time and Frequency Characterization cos/3coscos3x tttt cos1/3cos1cos31y tttt jHje Hj 1Hj cos/31/3cos1cos33y tttt6.2.1 Line
3、ar and Nonlinear Phase1.Linear Phase0 tjejH1jH0 tjH 0 YjXjt 5 Chapter 6 Time and Frequency Characterization cos/3/3cos/2cos32/3y tttt tx tyt0 cos/3/3cos/2cos32/3 3y tttt2.Nonlinear Phase 11jjjH1jHtg-jH-126 Chapter 6 Time and Frequency CharacterizationHomework:6.5 6.2376.5 Consider a continuous-time
4、ideal bandbass filter whose frequency response is Chapter 6 Problem Solutionelsewhere ,03 ,1ccjH(a)If is the impulse response of this filter,determine a function such that th tg tgttthcsin(b)As is increased,does the impulse response of the filterget more concentrated or less concentrated about the o
5、rigin?cSolution ttgc2cos(b)It will get more concentrated about the origin.(a)8 Chapter 6 Problem Solution6.23 Shown in Figure 6.23 is for a lowpass filter.Determine and sketch the impulse response of the filter for each of the following phase characteristics:(a)jH0jH01cjHcjHjH ttthcsinTjejHjH TtTtth
6、csinTjH(b),where T is a constant.90 2/0 2/-jH(c)Chapter 6 Problem Solution0 0 2/2/-jjeejHjH0jccjtjccettj22sintjccettj22sin012c2cF2sinttc th ttthc2/sin2210 Chapter 7 Sampling Chapter 7 Sampling11 Chapter 7 Sampling-T 0 T 2T t tx1 tx2 tx3nTxnTxnTx321 txtxtx32112 Chapter 7 Sampling7.1 The Sampling Theo
7、rem 7.1.1 Impulse-Train Sampling-3T -2T -T 0 T 2T 3T 4T t tx 0 x TxTx 2 tx txp nTttpn13 Chapter 7 SamplingSampling Theorem:Let be a band-limited signal withThen is uniquely determined by its samples if whereMjX ,0 tx tx,1,0,nnTxMs2Ts2 txp tx tx nTttpnjHMscM0TcjHc14 Chapter 7 Samplingtts cos cos 00te
8、dreconstruc When aliasing occurs,02 s000 ,2 sss 常数txrss 0 ,00，Homework:7.1 7.2 7.3 7.6 7.97.3 The Effect of Undersampling:Aliasing 欠采样欠采样 混叠混叠15 Chapter 7 Problem Solution7.1 A real-valued signal is known to be uniquely determined by its samples when the sampling frequency is .For what values of is
9、guaranteed to be zero?txjX000,10s 5000 ,0jX16 Chapter 7 Problem Solution7.2 A continuous-time signal is obtained at the output of an ideal lowpass filter with cutoff frequency .If impulse-train sampling is performed on ,which of the following sampling periods would guarantee that can be recovered fr
10、om its sampled version using an appropriate lowpass filter?(a)T=0.510-3 (b)T=210-3 (c)T=10-4 tx000,1c tx tx 000,1cM 000,22Ms(a)and(c)3max102sT Sampling interval17 Chapter 7 Problem Solution7.3 Determine the Nyquist rate corresponding to each of the following signals:t,t,tx 0004sin0002cos1 a tt,tx 00
11、04sin b 2 0004sin ctt,tx 000,4M 000,82Ms 000,4M 000,82Ms 000,8M 000,162Ms18 Chapter 7 Problem Solution7.6 11 ,0 jX twp tw tx1 tx2 tw nTttpn twp22 ,0 jXDetermine the maximum sampling interval T such that is recoverable from through the use of an ideal LPF.21 ,0 jWNyquist rate212 smaximum sampling int
12、erval21max2 sT jXjXjW212119 Chapter 7 Problem Solution7.9 Consider the signal 2 50sintttxwhich we wish to sample with a sampling frequency ofto obtain a signal with Fourier transform .Determinethe maximum value of for which it is guaranteed that150s0 75 jXjG tgjG0100050jX100050jG100100150150100 5002
13、0 Chapter 8 Communication Systems21 Chapter 8 Communication SystemsModulation:The general process of embedding an information-bearing signal into a second signal.Extracting the information-bearing signal.Demodulation:Modulation:Amplitude Modulation(AM)Frequency Modulation(FM)Phase Modulation(PM)Modu
14、latingSignal Carrier Signal ModulatedSignal tctxty tx tcAmplitude Modulation22 Chapter 8 Communication Systems8.1 Complex Exponential and Sinusoidal Amplitude Modulation8.1.1 Amplitude Modulation with a Complex Exponential Carrier cctjetcccarrier frequencycinitial phasecjXjYjX0MMAjY0MccAMc tjcetxtyM
15、odulation:Demodulation:tjcetytx23 Chapter 8 Communication Systems8.1.2 Amplitude Modulation with a Sinusoidal Carrier ccttc cos ttxtyccosModulation:ccjXjXjY2121jX0MMAjY0cc2AMcccctFcos24t txt tyDemodulationSynchronous Demodulation（同步解调）（同步解调）Asynchronous Demodulation（非同步解调）（非同步解调）Chapter 8 Communicat
16、ion SystemsEnvelope 25 Chapter 8 Communication Systemscctcos tx tyCommunicationChannel 8.2.1 Synchronous Demodulation Transmitter Modulation ReceiverSynchronous Demodulation ttxtwc2cos ttxtxc2cos2121LPF tx twcctcossamecc,jW0c2c22A4A4AMcLPF26 Chapter 8 Communication Systems1.Suppose the modulator and
17、 demodulator are not synchronized in phase.cctjeTransmitter Carrier Complex Exponential CarriercctjeReceiverCarrier cccctjtjeetxtw ccjetxtw ccWhen ,txtwrecover27 Chapter 8 Communication Systems Sinusoidal CarriercctcosTransmitter CarrierReceiverCarriercctcos cccctttxtwcoscosModulationDemodulation cc
18、cccttxtxtw2cos21cos21The output of LPF is:ccrtxtxcos21 txtxrcc 0 2txrccBe out of the passband of LPF28 Chapter 8 Communication Systems2.Suppose the modulator and demodulator are not synchronized in frequency.tccosTransmitter CarrierReceiverCarriertdcos tttxtwdc coscos ttxttxtwdcdc cos21cos21The outp
19、ut of LPF is:ttxtxdcr cos Be out of the passband of LPF txtxrdc txtxrdc 29 Chapter 8 Communication Systems ttxtxr cos dc M01jXM022jXrMMMM308.3 Frequency-Division Multiplexing（频分复用）（频分复用）Chapter 8 Communication Systemst1cos tx1t2cos tx2 ty1BPF1 ty2BPF2tncos txn tynBPF tynjX1M0MM0MjX2M0MjXn22110jYnn31
20、 Chapter 8 Communication Systemst1cos tx1t2cosBPF12tncos tynReceiver LPFM tx2BPFLPFM txnBPFLPFM32 Chapter 8 Communication SystemsHomework:8.1 8.3 8.2233 Chapter 8 Problem Solution8.1 MjX ,0cjjXjY 2Determine a signal such that tm tmtytx tjcetm 21Solution 34 Chapter 8 Problem Solution tm 0ty ttxttgtm
21、000,4sin21000,2cos 000,2 0 000,2 2 jH8.3 Determine .000,2 ,0jX ttxtg000,2sint2000cos tg tyLPF jH tySolution Be out of the passband of LPF35 8.22In Figure(a),a system is shown with input and outputThe input signal has the Fourier transform shown in Figure(b)Determine and sketch .Chapter 8 Problem Sol
22、utionjXjY tx ty01W3W3jH2 tx tr1 tr2W3W31jH1W5W5 ty tr3Wt3cosWt5cosjXW20W21Figure(a)Figure(b)36-7W -5W -3W 0 3W 5W 7W jR121-5W -3W 0 3W 5W jR221-8W -6W -2W 0 2W 6W 8W jR341-2W 0 2W jY41 Chapter 8 Problem Solution37Problems for Fourier Analysis tf 201t tf 2012t2 22 tttftf222 jejF tf2011ttsinA period o
23、f Example 1 Determine the Fourier transform of tftcostsin2jFj2jF2je238Example 2 A real continuous-time signal with Fourier transform ,and 1.If is even,determine .2.If is odd,determine .tf tf tf tf tfjFjFlnProblems for Fourier AnalysisejFjFln1.If is real,even tfejF212 Fte 2122etF 112ttf39Problems for
24、 Fourier Analysis2.If is real,odd tf sgnjejF ujeujejF jtueFt11uejtF211 uejtF211 jtjjtjtf1212 21 tttf40Problems for Fourier AnalysisExample 3 Consider the following LTI systems with impulse response:1.ttth4sin 28sin4sintttth2.tttth8cos4sin3.tttx6sin2cos tyIf the input is Determine the output 1.F4sint
25、tth1jH044 tty2cosSolution:42 jH41Problems for Fourier Analysis ttttth8sin4sin2.10441088214441212jH36 jH42 jH ttty6sin32cos4 tttth8cos4sin3.2/10441212jH2/16,02jHjH tty6sin2142Problems for Fourier AnalysisExample 5 In Figure(a),a system is shown with input signal and output signal .If the following in
26、formation are given.ttdtdthc2sin1cjejH/22 ttthc3sin3 tuth41.Determine jH12.Determine the impulse response of the whole system .th3.If the input signal ,determine the output signal 2/cos2sintttxcc ty th1 th4 th3jH2 ty tx+-ty tx43Problems for Fourier Analysis1.Determine jH1 ttdtdthc2sin1ccjjH 0 2 12cj
27、H10cc0cc22jH14402/1cc txjH2 ty+-Problems for Fourier AnalysisjHccj-HjH 0 121 2j-H21ccjc-e 0 121/2 ttthc2sinccctt/22/2sincccttt/2sin45Problems for Fourier Analysis3.If the input signal ,determine the output signal 2/cos2sintttxcc tyccjc-ejH 0 121/202cjH11212/jcejH11212/jcejH 2/costtyc46Problems for Fourier Analysis Consider an LTI system with unit impulse response sin2sintth tttt 2.If the input is determine the output 313sincoscos622x tttt y t1.Determine the frequency response of the whole system.
