DSP FIRST 2e 4. Sampling and Aliasing – Problems with selected Solutions 171


4–1  Sampling an AM signal  Solution  

  • 4–1
 two-sided spectrum   periodic   minimum sampling rate 
 

4–2  Reconstruction via realizable D-to-C  Solution  

  • 4–2
 pulse shape   rectangular   triangle   sample and hold 
 

4–3  Sampling and reconstruction of a chirp  Solution  

  • 4–3
 linear-FM   C-to-D   D-to-C   instantaneous frequency   specgram 
 

4–4  Spectrum of discrete-time signal and reconstruction of sinusoids  Solution  

  • 4–4
 aliasing   spectrum plot   complex amplitude 
 

4–5  Strobe sampling of rotating disk  Solution  

  • 4–5
 strobe   spectrum   aliasing 
 

4–6  C-to-D input derived from D-to-C output  Solution  

  • 4–6
 aliasing   C-to-D   D-to-C   sampling rate   reconstruction 
 

4–7  C-to-D input derived from D-to-C output 

  • 4–7
 aliasing   C-to-D   D-to-C   sampling rate   reconstruction 
 

4–8  C-to-D input derived from D-to-C output 

  • 4–8
 aliasing   C-to-D   D-to-C   sampling rate   reconstruction 
 

4–9  Sampling and Aliasing  Solution  

  • 4–9
 sampling   aliasing 
 

4–10  Sampling and Aliasing  Solution  

  • 4–10
 sampling   aliasing 
 

4–11  Sampling and Aliasing  Solution  

  • 4–11
 sampling   aliasing 
 

4–12  Sampling and Aliasing  Solution  

  • 4–12
 sampling   aliasing 
 

4–13  Sampling and Reconstruction of Cosine Signals  Solution  

  • 4–13
 sampling   reconstruction 
 

4–14  Sampling and Reconstruction of Cosine Signals  Solution  

  • 4–14
 sampling   reconstruction 
 

4–15  Sampling and Reconstruction of Cosine Signals  Solution  

  • 4–15
 sampling   reconstruction 
 

4–16  D/C Reconstruction for a Discrete-Time Chirp Signal  Solution  

  • 4–16
 chirp   converter   frequency   instantaneous 
 

4–17  Spectrum for AM Signal ♦ Minimum Sampling Rate  Solution  

  • 4–17
 AM   phasor   sampling   spectrum 
 

4–18  D/C Reconstruction for a Discrete-Time Chirp Signal  Solution  

  • 4–18
 chirp   converter   instantaneous   phasor 
 

4–19  Complex Roots of Unity for Complex Exponential Signal  Solution  

  • 4–19
 complex   exponential   roots 
 

4–20  C/D and D/C in Cascade  Solution  

  • 4–20
 cascade   discrete   sampling 
 

4–21  Aliased Discrete-Time Sinusoid Plot in MATLAB ♦ Sampling Theorem  Solution  

  • 4–21
 aliasing   MATLAB   sampling   stem 
 

4–22  Spectrum of AM Signal \(\sin(t)\)\(\cos(t)\) ♦ Minimum Sampling Rate  Solution  

  • 4–22
 AM   multiplication   sampling   spectrum 
 

4–23  Sampled Sinusoid ♦ Over-Sampled or Under-Sampled  Solution  

  • 4–23
 amplitude   phase   sampling   sinusoid 
 

4–24  Strobe Demo  Solution  

  • 4–24
 phasor   sampling   strobe 
 

4–25  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ D/C Reconstruction  Solution  

  • 4–25
 cascade   folding   sampling   spectrum 
 

4–26  Aliased Discrete-Time Sinusoid Plot in MATLAB ♦ Sampling Theorem  Solution  

  • 4–26
 MATLAB   period   sampling   stem 
 

4–27  C/D and D/C in Cascade ♦ Input Spectrum Given  Solution  

  • 4–27
 cascade   discrete   sampling   spectrum 
 

4–28  Spectrum Sinusoids ♦ Period & Minimum Sampling Rate  Solution  

  • 4–28
 period   sampling   sinusoid   spectrum 
 

4–29  D/C Reconstruction for a Discrete-Time Chirp Signal  Solution  

  • 4–29
 chirp   converter   instantaneous   phasor 
 

4–30  Spectrum Sinusoids ♦ Sampling ♦ Sketch Spectrum for \(x[n]\)  Solution  

  • 4–30
 phasor   sampling   spectrum 
 

4–31  C/D and D/C in Cascade ♦ Reconstruction of \(x[n]\)  Solution  

  • 4–31
 converter   output   reconstruction   sampling 
 

4–32  Strobe Demo  Solution  

  • 4–32
 phasor   sampling   spectrum   strobe 
 

4–33  C/D and D/C in Cascade ♦ Sketch Spectrum of \(x[n]\)  Solution  

  • 4–33
 cascade   converter   output   sampling   spectrum 
 

4–34  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ D/C Reconstruction  Solution  

  • 4–34
 converter   discrete   reconstruction   sampling 
 

4–35  Strobe Demo  Solution  

  • 4–35
 phasor   sampling   strobe 
 

4–36  D/C Reconstruction for a Given Reconstruction Pulse  Solution  

  • 4–36
 converter   output   reconstruction   sampling 
 

4–37  Sinusoids and Sampling  Solution  

  • 4–37
 period   complex exponential   amplitude   phase 
 

4–38  Sampling Theorem  Solution  

  • 4–38
 Nyquist Rate   reconstruction 
 

4–39  Discrete-time sinusoid from samples 

  • 4–39
 MATLAB   stem plot   phasor 
 

4–40  Sampling a Sinusoid 

  • 4–40
 sampling period 
 

4–41  Sampling and Aliasing a Sinusoid 

  • 4–41
 sampling frequency   oversampling   undersampling 
 

4–42  Sampling Rate from Spectrum of AM Sinusoid 

  • 4–42
 AM   phasor   sampling   spectrum 
 

4–43  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) 

  • 4–43
 amplitude   phase   sampling   continuous 
 

4–44  Strobe Demo 

  • 4–44
 rate   speed   strobe   aliasing   spectrum 
 

4–45  Sampling a Digital Chirp Signal 

  • 4–45
 phasor   D-A converter   aliasing 
 

4–46  Non-Ideal Reconstruction of Sampled Signals 

  • 4–46
 reconstruction   d-to-C converter 
 

4–47  TV and Rotating Wagon Wheel 

  • 4–47
 sampling rate   aliasing 
 

4–48  Sampling of Signals given the Spectrum  Solution  

  • 4–48
 sampling   cosine   aliasing 
 

4–49  Sampling and Reconstruction of a Chirp Signal  Solution  

  • 4–49
 Nyquist   instantaneous frequency 
 

4–50  Sampling of Signals given the Spectrum  Solution  

  • 4–50
 sampling   cosine   aliasing 
 

4–51  Sampling and Reconstruction of a Chirp Signal  Solution  

  • 4–51
 Nyquist   instantaneous frequency 
 

4–52  Sampling an AM signal  Solution  

  • 4–52
 spectrum   periodic   minimum sampling rate 
 

4–53  Sampling and reconstruction of a chirp  Solution  

  • 4–53
 linear-FM   C-to-D   D-to-C   instantaneous frequency   specgram 
 

4–54  Sampling and aliasing of sinusoids  Solution  

  • 4–54
 C-to-D   D-to-C   sampling rate   discrete-time spectrum 
 

4–55  Sampling and aliasing of sinusoids  Solution  

  • 4–55
 C-to-D   D-to-C   sampling rate   discrete-time spectrum 
 

4–56  Sampling Cosine Signals  Solution  

  • 4–56
 Sampling   Nyquist Rate 
 

4–57  Sampling Cosine Signals  Solution  

  • 4–57
 Sampling   Nyquist Rate 
 

4–58  Sampling Cosine Signals  Solution  

  • 4–58
 Sampling   Nyquist Rate 
 

4–59  Sampling a Bandlimited Periodic Signal  Solution  

  • 4–59
 Sampling   Periodic Signal 
 

4–60  Strobe Sampling of a Rotating Disk  Solution  

  • 4–60
 Strobe   Rotating Disk 
 

4–61  Sampling a Cosine Signal using MATLAB  Solution  

  • 4–61
 MATLAB   Spectrum   Sampling 
 

4–62  Sampling a Cosine Signal using MATLAB  Solution  

  • 4–62
 MATLAB   Spectrum   Sampling 
 

4–63  Non-Ideal Reconstruction of Sampled Signals  Solution  

  • 4–63
 Reconstruction 
 

4–64  Sampling and Reconstruction of a Chirp Signal  Solution  

  • 4–64
 Sampling   Reconstruction   Chirp 
 

4–65  Sampling of Cosine Signals  Solution  

  • 4–65
 Sampling   Cosine   Aliasing 
 

4–66  Sampling and Reconstruction of a Chirp Signal  Solution  

  • 4–66
 Sampling   Reconstruction   Chirp 
 

4–67  Sampling of Cosine Signals  Solution  

  • 4–67
 Sampling   Cosine   Aliasing 
 

4–68  Sampling and Reconstruction of a Chirp Signal  Solution  

  • 4–68
 Sampling   Reconstruction   Chirp 
 

4–69  Sampling of Cosine Signals  Solution  

  • 4–69
 Sampling   Cosine   Aliasing 
 

4–70  Sampling a Bandlimited Periodic Signal 

  • 4–70
 sampling   spectrum   reconstruction 
 

4–71  Sampling a Bandlimited Periodic Signal 

  • 4–71
 sampling   spectrum   reconstruction 
 

4–72  Sampling a Bandlimited Periodic Signal 

  • 4–72
 sampling   spectrum   reconstruction 
 

4–73  Non-Ideal Reconstruction of Sampled Signals 

  • 4–73
 nonideal reconstruction   interpolation 
 

4–74  Strobe Sampling of a Rotating Disk 

  • 4–74
 Strobe   Rotating Disk 
 

4–75  Sampling and Reconstruction of a Chirp Signal 

  • 4–75
 sampling   reconstruction   chip 
 

4–76  Sampling a Bandlimited Periodic Signal  Solution  

  • 4–76
 sampling   bandlimited 
 

4–77  Sampling a Periodic C-T Signal to Obtain a Periodic D-T Signal  Solution  

  • 4–77
 Sampling   Periodic Signal 
 

4–78  Sampling of Cosine Signals  Solution  

  • 4–78
 sampling   aliasing   reconstruction 
 

4–79  Sampling of Cosine Signals  Solution  

  • 4–79
 sampling   aliasing   reconstruction 
 

4–80  Sampling of Cosine Signals  Solution  

  • 4–80
 sampling   aliasing   reconstruction 
 

4–81  Sampling of AM signal from its spectrum  Solution  

  • 4–81
 Nyquist rate   minimum sampling rate   discrete-time spectrum 
 

4–82  Sampling & aliasing given continuous-time spectrum  Solution  

  • 4–82
 Discrete-time spectrum   minimum sampling rate 
 

4–83  Reconstruction from discrete-time spectrum  Solution  

  • 4–83
 sampling rate   aliasing   Nyquist rate   D-to-C converter 
 

4–84  Reconstruction via realizable D-to-C  Solution  

  • 4–84
 pulse shape   rectangular   triangle   sample and hold 
 

4–85  Strobe sampling of rotating disk  Solution  

  • 4–85
 strobe   spectrum   aliasing 
 

4–86  Instantaneous frequency of chirp after sampling & reconstruction  Solution  

  • 4–86
 folding   sampling rate   specgram 
 

4–87  Determine sampling rate from input and sampled sinusoids  Solution  

  • 4–87
 C-to-D converter   D-to-C   discrete-time spectrum 
 

4–88  Sampling & reconstruction of periodic signal  Solution  

  • 4–88
 C-to-D converter   D-to-C   discrete-time spectrum   period   fundamental frequency 
 

4–89  Period of discrete-time signal when sampling  Solution  

  • 4–89
 sampling rate   sinusoid   Nyquist rate 
 

4–90  File sizes for music representations  Solution  

  • 4–90
 synthetic music   sheet music   D-to-A converter 
 

4–91  Sampling and reconstruction of sum of sinusoids  Solution  

  • 4–91
 sampling rate   Nyquist rate   discrete-time spectrum   DC value 
 

4–92  Sampling and reconstruction of sum of sinusoids  Solution  

  • 4–92
 sampling rate   Nyquist rate   discrete-time spectrum   DC value 
 

4–93  Sampling and reconstruction of sum of sinusoids  Solution  

  • 4–93
 sampling rate   Nyquist rate   discrete-time spectrum   DC value 
 

4–94  Strobe Demo  Solution  

  • 4–94
 rate   speed   strobe 
 

4–95  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ Sampling Theorem  Solution  

  • 4–95
 amplitude   phase   sampling   continuous 
 

4–96  Sampled Sinusoid ♦ Over-Sampled or Under-Sampled  Solution  

  • 4–96
 amplitude   frequency   phase   sampling 
 

4–97  C/D and D/C in Cascade  Solution  

  • 4–97
 cascade   converter   output   sampling 
 

4–98  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ Sampling Theorem  Solution  

  • 4–98
 converter   sampling   sinusoid   continuous 
 

4–99  Complex Roots of Unity for Complex Exponential Signal  Solution  

  • 4–99
 complex   exponential   roots 
 

4–100  Sampled Sinusoid ♦ Over-Sampled or Under-Sampled  Solution  

  • 4–100
 amplitude   frequency   phase   sampling 
 

4–101  Aliased Discrete-Time Sinusoid Plot in MATLAB ♦ Sampling Theorem  Solution  

  • 4–101
 aliasing   folding   MATLAB   sampling 
 

4–102  Strobe Demo  Solution  

  • 4–102
 rate   speed   strobe 
 

4–103  Spectrum of Discrete-Time Sinusoid Defined by MATLAB Code  Solution  

  • 4–103
 MATLAB   phasor   spectrum 
 

4–104  D/C Reconstruction for a Given Reconstruction Pulse  Solution  

  • 4–104
 converter   period   reconstruction   sampling 
 

4–105  Spectrum Sinusoids ♦ Sampling ♦ Sketch Spectrum for \(x[n]\)  Solution  

  • 4–105
 phasor   sampling   spectrum 
 

4–106  Strobe Demo: Wagon Wheel  Solution  

  • 4–106
 rate   speed   strobe 
 

4–107  Sampling Theorem  Solution  

  • 4–107
 Nyquist Rate   reconstruction   MATLAB   soundsc   C-to-D converter 
 

4–108  Sampling of Signals Given the Spectrum 

  • 4–108
 sampling   cosine   aliasing   C-to-D   D-to-C 
 

4–109  Strobe Demo 

  • 4–109
 rate   speed   strobe   aliasing   spectrum 
 

4–110  Non-Ideal Reconstruction of Sampled Signals 

  • 4–110
 reconstruction   D-to-C converter 
 

4–111  Sampling of Signals Given the Spectrum  Solution  

  • 4–111
 sampling   cosine   frequency   amplitude   phase 
 

4–112  Sampling & Reconstruction of Sinusoids  Solution  

  • 4–112
 aliasing   sampling rate   spectrum   C-to-D converter   D-to-C converter 
 

4–113  Aliased Discrete-Time Sinusoid Plot in MATLAB ♦ Sampling Theorem  Solution  

  • 4–113
 aliasing   MATLAB   sampling   sinusoid 
 

4–114  Strobe Demo  Solution  

  • 4–114
 rate   speed   strobe 
 

4–115  D/C Reconstruction for a Discrete-Time Chirp Signal  Solution  

  • 4–115
 chirp   converter   phasor   reconstruction   rotating 
 

4–116  Sampled Sinusoid ♦ Over-Sampled or Under-Sampled  Solution  

  • 4–116
 amplitude   phase   sampling 
 

4–117  Strobe Demo  Solution  

  • 4–117
 rate   speed   strobe 
 

4–118  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ Sampling Theorem  Solution  

  • 4–118
 analog   sampling   continuous 
 

4–119  Strobe Demo  Solution  

  • 4–119
 rate   speed   strobe 
 

4–120  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ Sampling Theorem  Solution  

  • 4–120
 analog   sampling   continuous 
 

4–121  Discrete-Time Signal \(x[n]\) Derived from Sampling an Input Spectrum  Solution  

  • 4–121
 discrete   sampling   spectrum 
 

4–122  C/D and D/C in Cascade ♦ Determine Inputs from Output  Solution  

  • 4–122
 cascade   converter   sampling   continuous 
 

4–123  Strobe Demo  Solution  

  • 4–123
 phasor   spectrum   strobe 
 

4–124  Spectrum Sinusoids ♦ Sketch Spectrum for Sampled Signal  Solution  

  • 4–124
 phasor   rotating   sampling   spectrum 
 

4–125  C/D and D/C in Cascade ♦ Choosing Sampling Frequency  Solution  

  • 4–125
 cascade   converter   frequency   sampling 
 

4–126  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ D/C Reconstruction  Solution  

  • 4–126
 converter   reconstruction   sampling   continuous 
 

4–127  Strobe Demo  Solution  

  • 4–127
 phasor   spectrum   strobe 
 

4–128  Aliased Discrete-Time Sinusoid Plot in MATLAB ♦ Sampling Theorem  Solution  

  • 4–128
 aliasing   discrete   folding   MATLAB   sampling 
 

4–129  Strobe Demo  Solution  

  • 4–129
 rate   speed   strobe 
 

4–130  Strobe Demo  Solution  

  • 4–130
 rate   speed   strobe 
 

4–131  Line Spectrum of a Periodic Signal ♦ Minimum Sampling Rate  Solution  

  • 4–131
 multiplication   period   sampling   spectrum 
 

4–132  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ Sampling Theorem  Solution  

  • 4–132
 amplitude   discrete   phase   sampling   continuous 
 

4–133  Sampled Sinusoid ♦ Over-Sampled or Under-Sampled  Solution  

  • 4–133
 amplitude   phase   sampling 
 

4–134  C/D and D/C in Cascade  Solution  

  • 4–134
 cascade   converter   sampling   sum 
 

4–135  Strobe Demo  Solution  

  • 4–135
 rate   speed   strobe 
 

4–136  Continuous-Time Sinusoid From Discrete-Time \(x[n]\) ♦ Sampling Theorem  Solution  

  • 4–136
 converter   discrete   sampling   continuous 
 

4–137  Complex Roots of Unity for Complex Exponential Signal  Solution  

  • 4–137
 complex   exponential   polynomial   roots 
 

4–138  Strobe Demo  Solution  

  • 4–138
 rate   speed   strobe 
 

4–139  Aliased Discrete-Time Sinusoid Plot in MATLAB  Solution  

  • 4–139
 aliasing   discrete   MATLAB   sinusoid 
 

4–140  C/D and D/C in Cascade ♦ Input Spectrum Given  Solution  

  • 4–140
 cascade   converter   sampling   spectrum 
 

4–141  Strobe Demo 

  • 4–141
 complex phasor 
 

4–142  Sinusoid and Chirp through D/A converter 

  • 4–142
 aliasing   sampling rate 
 

4–143  D/C Reconstruction for a Discrete-Time Chirp Signal  Solution  

  • 4–143
 chirp   converter   discrete   phasor   reconstruction 
 

4–144  Complex Roots of Unity for Complex Exponential Signal  Solution  

  • 4–144
 complex   exponential   polynomial   roots 
 

4–145  C/D and D/C in Cascade  Solution  

  • 4–145
 cascade   converter   sampling 
 

4–146  Aliased Discrete-Time Sinusoid Plot in MATLAB ♦ Sampling Theorem  Solution  

  • 4–146
 aliasing   discrete   folding   MATLAB   sampling   stem 
 

4–147  Strobe Demo  Solution  

  • 4–147
 rate   speed   strobe 
 

4–148  Strobe Demo: Wagon Wheel  Solution  

  • 4–148
 rate   speed   strobe 
 

4–149  Discrete-time sinusoid from samples 

  • 4–149
 oversampled   undersampled 
 

4–150  Sampling and Reconstruction from Input Spectrum 

  • 4–150
 aliasing   sampling rate   spectrum   C-to-D converter   D-to-C converter 
 

4–151  D/C Reconstruction for Different Reconstruction Pulses  Solution  

  • 4–151
 converter   output   pulse   reconstruction 
 

4–152  Spectrum Sinusoids ♦ Minimum Sampling Rate  Solution  

  • 4–152
 period   sampling   sinusoid   spectrum 
 

4–153  Spectrum of Discrete-Time Sinusoid Defined by MATLAB Code  Solution  

  • 4–153
 discrete   MATLAB   phasor   sinusoid   spectrum 
 

4–154  C/D and D/C in Cascade ♦ Input Signal is a Chirp  Solution  

  • 4–154
 cascade   chirp   converter   sampling 
 

4–155  Strobe Demo ♦ Wagon Wheel  Solution  

  • 4–155
 phasor   rate   strobe 
 

4–156  Sampling Theorem  Solution  

  • 4–156
 Nyquist Rate   reconstruction   MATLAB   soundsc   C-to-D converter 
 

4–157  Sampling Theorem  Solution  

  • 4–157
 Nyquist Rate   reconstruction   MATLAB   soundsc   C-to-D converter 
 

4–158  Plot Spectrum of Summed Sinusoids Using MATLAB  Solution  

  • 4–158
 discrete-time spectrum 
 

4–159  Non-Ideal Reconstruction of Sampled Signals 

  • 4–159
 reconstruction   D-to-C converter 
 

4–160  Sampling of Signals Given the Spectrum  Solution  

  • 4–160
 sampling   cosine   aliasing 
 

4–161  Sampling of Signals Given the Spectrum  Solution  

  • 4–161
 sampling   cosine   aliasing 
 

4x–162  Discrete-Time Sinusoid. Derived by Sampling 

  • 4–162
 oversampled   undersampled 
 

4.1–163  MATLAB Code for Aliased Sinusoid  Solution  

  • 4–163
 aliasing   MATLAB   plot   sinusoid 
 

4.9–164  Response of FIR to Complex Exponential  Solution  

  • 4–164
 amplitude   difference   exponential   phase 
 

4.11–165  Sampling an AM Spectrum  Solution  

  • 4–165
 AM   multiplication   sampling   spectrum 
 

4.12–166  Sampling a Line Spectrum  Solution  

  • 4–166
 power   sampling   spectrum 
 

4.13–167  MATLAB Code for Aliased Sinusoid  Solution  

  • 4–167
 aliasing   folding   MATLAB   sinusoid 
 

4.14–168  Spectrum of AM Sinusoid  Solution  

  • 4–168
 AM   phasor   sampling   spectrum 
 

4.15–169  Cascade of A/D & D/A with Spectrum  Solution  

  • 4–169
 cascade   converter   sampling   spectrum 
 

4.16–170  Strobe Demo  Solution  

  • 4–170
 rate   speed   strobe 
 

4.18–171  Quadratic Phase  Solution  

  • 4–171
 chirp   exponential   phasor   quadratic