DSP FIRST 2e 3. Spectrum Representation – Problems with selected Solutions 165


3–1  Spectrum representation of sum of cosines  Solution  

  • 3–1
 line spectrum   complex amplitude   negative frequencies 
 

3–2  Spectrum representation of AM radio signal  Solution  

  • 3–2
 carrier signal 
 

3–3  Spectrum of sinusoid plus DC  Solution  

  • 3–3
 negative frequency   two-sided spectrum 
 

3–4  Matching instantaneous frequency to sinusoids  Solution  

  • 3–4
 chirp   instantaneous frequency   linear-FM 
 

3–5  Symmetry in the spectrum  Solution  

  • 3–5
 conjugate symmetry   complex amplitude   time shift   frequency 
 

3–6  Spectrum values from time-domain plot  Solution  

  • 3–6
 complex amplitude   frequency   amplitude   phase 
 

3–7  Sum of sinusoids from spectrum  Solution  

  • 3–7
 complex amplitude   periodic   fundamental frequency 
 

3–8  Symmetry in the spectrum 

  • 3–8
 conjugate symmetry   complex amplitude   time shift   frequency 
 

3–9  Spectrum values from time-domain plot 

  • 3–9
 complex amplitude   frequency   amplitude   phase 
 

3–10  Sum of sinusoids from spectrum 

  • 3–10
 complex amplitude   periodic   fundamental frequency 
 

3–11  Symmetry in the spectrum 

  • 3–11
 conjugate symmetry   complex amplitude   time shift   frequency 
 

3–12  Spectrum values from time-domain plot 

  • 3–12
 complex amplitude   frequency   amplitude   phase 
 

3–13  Sum of sinusoids from spectrum 

  • 3–13
 complex amplitude   periodic   fundamental frequency 
 

3–14  Symmetry in the spectrum 

  • 3–14
 conjugate symmetry   complex amplitude   time shift   frequency 
 

3–15  Spectrum values from time-domain plot 

  • 3–15
 complex amplitude   frequency   amplitude   phase 
 

3–16  Sum of sinusoids from spectrum 

  • 3–16
 complex amplitude   periodic   fundamental frequency 
 

3–17  Symmetry in the spectrum 

  • 3–17
 conjugate symmetry   complex amplitude   time shift   frequency 
 

3–18  Spectrum values from time-domain plot 

  • 3–18
 complex amplitude   frequency   amplitude   phase 
 

3–19  Sum of sinusoids from spectrum 

  • 3–19
 complex amplitude   periodic   fundamental frequency 
 

3–20  Fourier series coefficients of sum of sinusoids  Solution  

  • 3–20
 period 
 

3–21  Fourier series coefficients of sum of sinusoids 

  • 3–21
 period 
 

3–22  Fourier series coefficients of sum of sinusoids 

  • 3–22
 period 
 

3–23  Determine Signal from Incomplete Spectrum Plot  Solution  

  • 3–23
 cosine signal   spectrum 
 

3–24  Spectrum of AM Modulated Signal  Solution  

  • 3–24
 AM Modulation   spectrum 
 

3–25  Determine Signal from Incomplete Spectrum Plot  Solution  

  • 3–25
 cosine signal   spectrum 
 

3–26  Spectrum of AM Modulated Signal  Solution  

  • 3–26
 AM Modulation   spectrum 
 

3–27  Determine Signal from Incomplete Spectrum Plot  Solution  

  • 3–27
 cosine signal   spectrum 
 

3–28  Spectrum of AM Modulated Signal  Solution  

  • 3–28
 AM Modulation   spectrum 
 

3–29  Determine Signal from Incomplete Spectrum Plot  Solution  

  • 3–29
 cosine signal   spectrum 
 

3–30  Spectrum of AM Modulated Signal  Solution  

  • 3–30
 AM Modulation   spectrum 
 

3–31  Determine Signal from Incomplete Spectrum Plot  Solution  

  • 3–31
 cosine signal   spectrum 
 

3–32  Spectrum of AM Modulated Signal  Solution  

  • 3–32
 AM Modulation   spectrum 
 

3–33  Determine Signal from Incomplete Spectrum Plot  Solution  

  • 3–33
 cosine signal   spectrum 
 

3–34  Spectrum of AM Modulated Signal  Solution  

  • 3–34
 AM Modulation   spectrum 
 

3–35  Determine Fourier Series for a Sum of Cosine Signals  Solution  

  • 3–35
 Fourier Series   cosine signals 
 

3–36  Determine Fourier Series for a Sum of Cosine Signals  Solution  

  • 3–36
 Fourier Series   cosine signals 
 

3–37  Determine Fourier Series for a Sum of Cosine Signals  Solution  

  • 3–37
 Fourier Series   cosine signals 
 

3–38  Spectrum Sinusoids ♦ Period  Solution  

  • 3–38
 period   sinusoid   spectrum 
 

3–39  Spectrum for Sine Cubed ♦ Fundamental Period  Solution  

  • 3–39
 fundamental   period   spectrum   sum 
 

3–40  Sinusoids Defined by MATLAB Spectrum  Solution  

  • 3–40
 MATLAB   multiplication   spectrum 
 

3–41  Spectrum Complex Exponentials ♦ Period  Solution  

  • 3–41
 complex   exponential   period   phasor   spectrum   sum 
 

3–42  Spectrum sum of sinusoids 

  • 3–42
 period   sinusoid   spectrum 
 

3–43  Plot an Amplitude Modulated Signal 

  • 3–43
 AM   MATLAB   spectrum   plot 
 

3–44  Compute Frequencies of Notes in the C-Major Scale 

  • 3–44
 music   scale   frequency   note 
 

3–45  Spectrum of a Sum of Cosine Signals 

  • 3–45
 spectrum   cosine   periodicity 
 

3–46  Multiplication of Two Sinusoids 

  • 3–46
 sinusoids   cosines   period 
 

3–47  Fourier Series Analysis of a Periodic Square Wave 

  • 3–47
 DC component   Fourier coefficients   analysis integral 
 

3–48  Instantaneous Frequency of a Chirp Signal 

  • 3–48
 linear   FM 
 

3–49  Features of Pure Sinusoids  Solution  

  • 3–49
 fundamental period   frequency   amplitude   spectrum 
 

3–50  Match Spectrum to Periodic Waveforms  Solution  

  • 3–50
 spectrum   cosine   match   period 
 

3–51  Features of Pure Sinusoids  Solution  

  • 3–51
 fundamental period   frequency   amplitude   spectrum 
 

3–52  Match Spectrum to Periodic Waveforms  Solution  

  • 3–52
 spectrum   cosine   match   period 
 

3–53  Write signal from a spectrum  Solution  

  • 3–53
 symmetry   sum of cosines   complex amplitude   frequency 
 

3–54  Spectrum representation of AM radio signal  Solution  

  • 3–54
 carrier signal 
 

3–55  Matching spectrum and periodic signals  Solution  

  • 3–55
 fundamental frequency   period   complex amplitude 
 

3–56  Spectrum of sinusoid plus DC  Solution  

  • 3–56
 negative frequency   two-sided spectrum 
 

3–57  Frequencies of musical notes  Solution  

  • 3–57
 chord   triad   A-440   piano   octave   middle-C 
 

3–58  Matching instantaneous frequency to sinusoids  Solution  

  • 3–58
 chirp   instantaneous frequency   linear-FM 
 

3–59  Spectrum of sum of sinusoids  Solution  

  • 3–59
 complex amplitude   fundamental frequency 
 

3–60  Fourier Series Integral for Specific Signal  Solution  

  • 3–60
 Fourier Series   Periodic Signal 
 

3–61  Fourier Series Integral for Specific Signal  Solution  

  • 3–61
 Fourier Series   Periodic Signal 
 

3–62  Fourier Series Integral for Specific Signal  Solution  

  • 3–62
 Fourier Series   Periodic Signal 
 

3–63  DC Component of a Sum of Cosines  Solution  

  • 3–63
 DC   Cosine 
 

3–64  Beat Notes  Solution  

  • 3–64
 Beat   Spectrum 
 

3–65  Match the Waveform with its Spectrum  Solution  

  • 3–65
 Waveform   Spectrum 
 

3–66  Compute the Frequencies of the Notes of the C-Major Scale  Solution  

  • 3–66
 Music   Scale   Frequency   Note 
 

3–67  Spectrum of a Periodic Signal from the Fourier Series  Solution  

  • 3–67
 Spectrum   Periodic Signal 
 

3–68  Compute the DC Component of a Periodic Signal  Solution  

  • 3–68
 DC   Periodic   Fourier Series 
 

3–69  Compute the Fourier Series Coefficients for a Periodic Pulse Signal  Solution  

  • 3–69
 Fourier Coefficients   Pulse Signal 
 

3–70  Determine the Instantaneous Frequency of a Chirp Signal  Solution  

  • 3–70
 Chirp   FM   Instantaneous Frequency 
 

3–71  Determine a Chirp Signal given the Instantaneous Frequency  Solution  

  • 3–71
 Chirp   FM   Instantaneous Frequency 
 

3–72  Match Spectrum to Sinusoidal Waveforms  Solution  

  • 3–72
 Spectrum   Cosine   Match 
 

3–73  Match Spectrum to Sinusoidal Waveforms  Solution  

  • 3–73
 Spectrum   Cosine   Match 
 

3–74  Match Spectrum to Sinusoidal Waveforms  Solution  

  • 3–74
 Spectrum   Cosine   Match 
 

3–75  Evaluate Fourier Series Integral for a Specific Signal  Solution  

  • 3–75
 Fourier Series   Periodic Signal 
 

3–76  Evaluate Fourier Series Integral for a Specific Signal  Solution  

  • 3–76
 Fourier Series   Periodic Signal 
 

3–77  Evaluate Fourier Series Integral for a Specific Signal  Solution  

  • 3–77
 Fourier Series   Periodic Signal 
 

3–78  Spectrum of a Sum of Cosine Signals 

  • 3–78
 spectrum   cosine signal 
 

3–79  DC Component of a Sum of Cosines 

  • 3–79
 DC   sinusoids 
 

3–80  Express the Square of a Sinusoid as a Sum of Complex Exponentials 

  • 3–80
 square   sinusoid   complex exponential 
 

3–81  Spectrum of a Sum of Cosine Signals 

  • 3–81
 spectrum   cosines 
 

3–82  Compute the Frequencies of the Notes of the C-Major Scale 

  • 3–82
 Music   Scale   Frequency   Note 
 

3–83  Determine the Instantaneous Frequency of a Chirp Signal 

  • 3–83
 Chirp   FM   Instantaneous Frequency 
 

3–84  Determine the Fourier Series of a Product of Sinusoids 

  • 3–84
 Fourier Series   Periodic Signal 
 

3–85  Fourier Series Integral for Specific Signal 

  • 3–85
 Fourier Series   Periodic Signal 
 

3–86  Spectrum of a Sum of Cosine Signals  Solution  

  • 3–86
 spectrum   cosine signal 
 

3–87  Spectrum of a Sum of Cosine Signals  Solution  

  • 3–87
 spectrum   cosine signal 
 

3–88  Spectrum of a Sum of Cosine Signals  Solution  

  • 3–88
 spectrum   cosine signal 
 

3–89  Evaluate Fourier Series Integral for a Specific Signal  Solution  

  • 3–89
 Fourier series   DC value   periodic signal 
 

3–90  Evaluate Fourier Series Integral for a Specific Signal  Solution  

  • 3–90
 Fourier series   DC value   periodic signal 
 

3–91  Evaluate Fourier Series Integral for a Specific Signal  Solution  

  • 3–91
 Fourier series   DC value   periodic signal 
 

3–92  Fourier series coefficients for sum of sinusoids  Solution  

  • 3–92
 period   complex amplitude   phasor addition 
 

3–93  Fourier series coefficients for sum of sinusoids  Solution  

  • 3–93
 period   complex amplitude   phasor addition 
 

3–94  Fourier series coefficients for sum of sinusoids  Solution  

  • 3–94
 period   complex amplitude   phasor addition 
 

3–95  Write sum of sinusoids from spectrum information  Solution  

  • 3–95
 two-sided spectrum   conjugate symmetry   complex phasor   complex amplitude 
 

3–96  Spectrum of cosine squared  Solution  

  • 3–96
 Euler's formula 
 

3–97  Spectrum of sinusoid plus DC  Solution  

  • 3–97
 complex amplitude 
 

3–98  Write sum of sinusoids from spectrum information  Solution  

  • 3–98
 two-sided spectrum   complex amplitude   time-shifted sinusoid 
 

3–99  Add sinusoid to an existing spectrum  Solution  

  • 3–99
 periodic   complex amplitude   fundamental frequency 
 

3–100  Frequencies of musical notes  Solution  

  • 3–100
 chord   triad   A-440   piano   octave   middle-C 
 

3–101  Matching spectra to periodic signals  Solution  

  • 3–101
 complex amplitude   DC value 
 

3–102  Instantaneous frequency of chirp signals  Solution  

  • 3–102
 linear-FM 
 

3–103  Plot periodic \(x(t)\) defined by Fourier integral  Solution  

  • 3–103
 Fourier series   DC value 
 

3–104  Derivative and time-shift of Fourier series representation  Solution  

  • 3–104
 properties of Fourier series 
 

3–105  Sum of sinusoids from two-sided spectrum plot  Solution  

  • 3–105
 complex amplitude   fundamental frequency 
 

3–106  Spectrum for product of sinusoids  Solution  

  • 3–106
 complex amplitude 
 

3–107  Sum of sinusoids from two-sided spectrum plot  Solution  

  • 3–107
 complex amplitude   fundamental frequency 
 

3–108  Spectrum for product of sinusoids  Solution  

  • 3–108
 complex amplitude 
 

3–109  Sum of sinusoids from two-sided spectrum plot  Solution  

  • 3–109
 complex amplitude   fundamental frequency 
 

3–110  Spectrum for product of sinusoids  Solution  

  • 3–110
 complex amplitude 
 

3–111  Fourier series and spectrum  Solution  

  • 3–111
 period 
 

3–112  Fourier series and spectrum  Solution  

  • 3–112
 period 
 

3–113  Fourier series and spectrum  Solution  

  • 3–113
 period 
 

3–114  Spectrum of \(\sin(t)\)\(\sin(t)\) Defined by MATLAB Code  Solution  

  • 3–114
 MATLAB   multiplication   spectrum 
 

3–115  Spectrum of \(\sin(t)\)+\(\sin(t)\) Defined by MATLAB Code  Solution  

  • 3–115
 MATLAB   spectrum   sum 
 

3–116  Sinusoids Spectrum ♦ Period  Solution  

  • 3–116
 exponential   period   phasor   spectrum 
 

3–117  Spectrum of \(\cos(t)\)\(\sin(t)\) Defined by MATLAB Code  Solution  

  • 3–117
 MATLAB   multiplication   spectrum 
 

3–118  Instantaneous Frequency of a Linear-FM (Chirp) Signal  Solution  

  • 3–118
 chirp   exponential   frequency   instantaneous 
 

3–119  Instantaneous Frequency of a Linear-FM (Chirp) Signal  Solution  

  • 3–119
 chirp   frequency   instantaneous   plot 
 

3–120  Sinusoids and Complex Numbers  Solution  

  • 3–120
 period   amplitude   phase   fundamental frequency 
 

3–121  Spectrum of a Sum of Cosine Signals  Solution  

  • 3–121
 spectrum   cosine   periodicity   complex amplitude 
 

3–122  Spectrum of AM Signal 

  • 3–122
 carrier signal   sinusoid   spectrum   frequency 
 

3–123  Spectrum of Signal Composed of Sinusoids 

  • 3–123
 Fourier Series   synthesis   fundamental period   DC 
 

3–124  Fourier Series Analysis and Spectrum of a Periodic Square Wave 

  • 3–124
 DC component   Fourier coefficients   analysis integral 
 

3–125  Match Spectrum to Periodic Waveforms 

  • 3–125
 complex amplitude 
 

3–126  Compute Frequencies of Notes in the C-Major Scale 

  • 3–126
 music   scale   frequency   note 
 

3–127  Plot Spectrum of Summed Sinusoids Using MATLAB 

  • 3–127
 discrete-time spectrum 
 

3–128  Instantaneous Frequency of a Chirp Signal 

  • 3–128
 linear   FM 
 

3–129  Match Spectrum to Periodic Waveforms  Solution  

  • 3–129
 complex amplitude 
 

3–130  Match Spectrum to Periodic Waveforms  Solution  

  • 3–130
 complex amplitude 
 

3–131  Symmetry of the Spectrum  Solution  

  • 3–131
 fundamental frequency   conjugate symmetry 
 

3–132  Relate Time Shift to Phase of a Sinusoid  Solution  

  • 3–132
 phase   fundamental period   maximum 
 

3–133  Fourier Integral & DC Value of Periodic Signal  Solution  

  • 3–133
 bandlimited   fundamental frequency   instantaneous frequency 
 

3–134  Instantaneous Frequency of a Linear-FM (Chirp) Signal  Solution  

  • 3–134
 chirp   exponential   frequency   instantaneous 
 

3–135  Instantaneous Frequency of a Linear-FM (Chirp) Signal  Solution  

  • 3–135
 chirp   frequency   instantaneous   sweep 
 

3–136  Sinusoids Spectrum ♦ Period  Solution  

  • 3–136
 period   phasor   spectrum   sum 
 

3–137  Instantaneous Frequency of a Linear-FM (Chirp) Signal  Solution  

  • 3–137
 chirp   derivative   frequency   instantaneous 
 

3–138  Spectrum Sinusoids ♦ Fundamental Frequency and Period  Solution  

  • 3–138
 frequency   fundamental   period   sinusoid   spectrum 
 

3–139  Spectrum for Sine Cubed  Solution  

  • 3–139
 exponential   fundamental   power 
 

3–140  Spectrum for AM Signal  Solution  

  • 3–140
 AM   phasor   spectrum 
 

3–141  Spectrum for AM Signal  Solution  

  • 3–141
 AM   phasor   spectrum 
 

3–142  Spectrum Complex Exponentials ♦ Period  Solution  

  • 3–142
 period   sampling   sinusoid   spectrum 
 

3–143  Spectrum of \(\cos(t)\)\(\cos(t)\) Defined by MATLAB Code  Solution  

  • 3–143
 MATLAB   multiplication   spectrum 
 

3–144  Piano Frequencies  Solution  

  • 3–144
 frequency   music   spectrum 
 

3–145  Match Chirp to Sinusoids 

  • 3–145
 instantaneous frequency 
 

3–146  Spectrum of a Sum of Cosine Signals  Solution  

  • 3–146
 spectrum   cosine   periodicity 
 

3–147  Spectrum of Product of 2 Sinusoids  Solution  

  • 3–147
 period 
 

3–148  Match Spectrum to Periodic Waveforms  Solution  

  • 3–148
 complex amplitude 
 

3–149  Compute Frequencies of Notes in the C-Major Scale  Solution  

  • 3–149
 music   scale   frequency   note 
 

3–150  Instantaneous Frequency of a Chirp Signal  Solution  

  • 3–150
 linear   FM 
 

3–151  Spectrum of Real-Valued Signal  Solution  

  • 3–151
 complex amplitude 
 

3–152  Features of Sinusoids  Solution  

  • 3–152
 period   frequency   spectrum 
 

3–153  Spectrum of Real-Valued Signal  Solution  

  • 3–153
 complex amplitude 
 

3–154  Features of Sinusoids  Solution  

  • 3–154
 period   frequency   spectrum 
 

3.3–155  Spectrum Sinusoids ♦ Period  Solution  

  • 3–155
 period   sinusoid   spectrum 
 

3.4–156  Spectrum of \(\sin(t)\)-cubed  Solution  

  • 3–156
 exponential   period   spectrum   sum 
 

3.6–157  Spectrum of AM Sinusoidal Signal  Solution  

  • 3–157
 AM   phasor   sinusoid   spectrum 
 

3.11–158  Instantaneous Frequency of a Linear-FM (Chirp) Signal  Solution  

  • 3–158
 chirp   frequency   instantaneous   time 
 

3.11–159  Instantaneous Frequency of a Linear-FM (Chirp) Signal  Solution  

  • 3–159
 chirp   frequency   instantaneous   time 
 

3.12–160  Instantaneous Frequency Compared via MATLAB  Solution  

  • 3–160
 chirp   instantaneous   MATLAB   sampling   sweep 
 

3.12–161  Fourier Series for a Square Wave and Modifications 

  • 3–161
 Fourier Series   square wave 
 

3.12–162  Instantaneous Frequency Compared via MATLAB  Solution  

  • 3–162
 chirp   derivative   frequency   instantaneous 
 

3.14–163  Effect of Time-Domain Modifications on the Fourier Series of a Periodic Signal 

  • 3–163
 Fourier Series   modifications 
 

3.15–164  Fourier Series of a Delayed Square Wave 

  • 3–164
 Fourier Series   delay   square wave 
 

3.19–165  Match the Waveform with its Spectrum 

  • 3–165
 match   sinusoidal waveform   spectrum