Exercises: Solutions and Hints

Chapter 1

  1. 1.1 upper I Subscript normal upper L Baseline left-parenthesis t right-parenthesis equals one half plus StartFraction 2 Over normal pi EndFraction sigma-summation Underscript p equals 1 Overscript 4 Endscripts StartFraction left-parenthesis negative 1 right-parenthesis Superscript p plus 1 Baseline Over 2 p minus 1 EndFraction cosine 2 pi left-parenthesis 2 p minus 1 right-parenthesis StartFraction t Over upper T EndFraction
  2. 1.2 s(nT) = sin (nπ + ϕ) = (−1)n sin ϕ.

    The possibility of reconstruction depends on ϕ. left-parenthesis phi equals StartFraction pi Over 2 EndFraction y e s semicolon phi equals 0 n o right-parenthesis.

  3. 1.3 upper H left-parenthesis StartFraction f Subscript s Baseline Over 2 EndFraction right-parenthesis equals StartFraction 2 StartRoot 2 EndRoot Over pi EndFraction left-parenthesis 0.92 d upper B right-parenthesis period
  4. 1.4 f2 < fs < 2f1.
  5. 1.5 s left-parenthesis italic n upper T right-parenthesis equals s Subscript r Baseline left-parenthesis italic n upper T right-parenthesis plus italic j s Subscript i Baseline left-parenthesis italic n upper T right-parenthesis equals e Superscript j left-parenthesis pi slash 2 right-parenthesis n Baseline StartStartFraction sine StartFraction 3 pi Over 8 EndFraction n OverOver sine left-parenthesis StartFraction pi Over 8 EndFraction n right-parenthesis EndEndFraction period
  6. 1.6 Maximum value of s(n) = 8;
    s left-parenthesis n right-parenthesis equals 0 for phi Subscript k Baseline equals minus 2 pi StartFraction k Over 8 EndFraction n plus k pi
  7. 1.7 ƒs = 2 MHz; Δƒ = 1 kHz.
  8. 1.8 p left-parenthesis 1 right-parenthesis equals StartFraction 1 Over pi EndFraction StartFraction 1 Over StartRoot upper A squared minus s squared EndRoot EndFraction semicolon r left-parenthesis tau right-parenthesis equals 2 left-parenthesis f 2 minus f 1 right-parenthesis StartFraction sine pi left-parenthesis f 2 minus f 1 right-parenthesis tau Over pi left-parenthesis f 2 minus f 1 right-parenthesis tau EndFraction cosine pi left-parenthesis f 2 plus f 1 right-parenthesis tau period
  9. 1.9 Periodic part; Fourier coefficient: upper C Subscript n Baseline equals StartStartFraction p sine StartFraction pi Over 2 EndFraction n OverOver pi n EndEndFraction period

    Non-periodic part; spectrum: normal upper S 2 left-parenthesis f right-parenthesis equals p left-parenthesis 1 minus p right-parenthesis normal upper T StartFraction 1 minus cosine pi f normal upper T Over pi squared f squared normal upper T squared EndFraction.

  10. 1.10 Signal-to-noise ratio in the band: 300–500 Hz = 75 dB (ƒs = 16 kHz; gain 3 dB).
  11. 1.11 Quantizing distortion: line at three eighths f Subscript normal s Baseline semicolon power: 0.01952.
  12. 1.12 If the characteristic is centered:
    a 1 equals 0 for 0 less-than-or-equal-to StartAbsoluteValue alpha EndAbsoluteValue less-than-or-equal-to one half semicolon a 1 equals StartFraction 4 q Over pi EndFraction StartRoot 1 minus StartFraction 1 Over 4 alpha squared EndFraction EndRoot for one half less-than-or-equal-to StartAbsoluteValue alpha EndAbsoluteValue less-than-or-equal-to 1

    Centering at ...

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