PROBLEMS
2.18 Bandwidth Requirements for Analog Information Signals
As Fourier methods show, signals can be described in terms of a frequency spectrum. From the transmission point of view, it is desirable to limit the bandwidth required for any signal, because this allows more channels to be accommodated in a given frequency band. Thus the bandwidth allocated for signals is often a com- promise between minimizing the bandwidth and acceptable levels of distortion.
For speech signals, listening tests have shown that the energy content resides mainly in the low audio- frequency range; typically, about 80% of the energy lies in frequencies below 1 kHz. The intelligibility, or clarity, of the signal requires the higher frequencies, typically in the range from 1.5 to 2.5 kHz. For telephone service, a frequency range of 300 to 3400 Hz is the generally accepted standard, even though the spectrum for natural speech has a much greater frequency spread than this. Not all telephone administrations adhere to this standard, but it is widely used in determining system performance. Thus the audio bandwidth for speech signals is 3400 – 300 3100 Hz. As will be seen later, when filtering requirements are taken into account, 4 kHz is usually allocated as the channel bandwidth for speech signals.
Signals associated with music, both instrumental and voice, require a much larger bandwidth than that for speech. Listening tests have shown that a frequency range of about 15 to 20,000 Hz is required for reason- able high-fidelity sound. For example, the specification sheets for certain pieces of “compatible” high-fidelity equipment specify the frequency ranges as CD player, 2 to 20,000 Hz; FM tuner section, 30 to 15,000 Hz;
stereo tape deck, normal tape, 30 to 14,000 Hz, metal and chrome tapes, 30 to 15,000 Hz; and stereo turntable, 10 to 30,000 Hz.
Video signals, such as those produced by standard television systems, require a bandwidth of about 4 MHz, while a facsimile signal requires a bandwidth of only about 1000 Hz. These bandwidth requirements are discussed in detail in later chapters, but for the present they are given here for comparison purposes. This wide difference in bandwidth requirements reflects the difference in time taken for a transmission system to scan the picture information. With television the picture is scanned in about 1/30 s, while a facsimile scan- ner may require about 10 min for a page of print. This illustrates the general connection between speed and bandwidth. The greater the rate at which information is generated, the greater the bandwidth needed, and of course vice versa. Thus the widespread use of facsimile transmission has come about (apart from its general appeal to the public) because it makes use of existing telephone networks that are often limited in bandwidth.
These bandwidth requirements are shown in Fig. 2.18.1 on a logarithmic frequency scale. The video signal extends down to zero frequency (which cannot be shown on the logarithmic scale; why?). The point
G (f)
Tb f Tb
Tb
0 1 2 3
Figure 2.17.2 Power spectral density for a binary random waveform of rectangular pulses of amplitudes A.
has been made elsewhere (source unknown) that the video bandwidth is about 1000 times the speech band- width, which gives some credence to the saying that “one picture is worth a thousand words”!
Digital signals also follow the general principle that the bandwidth requirements increase as the trans- mission rate increases. Digital signals are covered in the next chapter.
PROBLEMS
2.1. A sine wave is described by 5 sin(300t27˚), where tis time in seconds. Determine the waveform (a) amplitude, (b) rms value, (c) frequency, (d) periodic time, and (e) time lag or lead.
2.2. Express the waveform of Problem 2.1 as a cosine function.
2.3. By sketching the waveform of Problem 2.1, determine if it is an even or odd function, or neither.
2.4. A periodic waveform is described by By sketching this function, determine if it is an even or odd function, or neither.
2.5. Repeat Problem 2.4 for f(x) xfor x .
2.6. The trigonometric series for a periodic waveform consists of three sine terms consisting of a fundamental, a second harmonic, and a third harmonic. The fundamental has an amplitude of 1 V and provides the reference phase. The second harmonic has an amplitude of 0.3 V and a phase lag of 27˚.
The third harmonic has an amplitude of 0.5 V and a phase lead of 30˚. Draw accurately to scale the resultant waveform.
2.7. The spectrum components of a waveform are fundamental, 1 V; second harmonic, 0.7 V; third harmonic,0.35 V; all are sine waves with zero phase angle. The fourth harmonic is a 0.1-V cosine wave, also with zero phase angle. Construct accurately to scale one cycle of the resultant waveform.
All voltages given are peak values.
2.8. A waveform consists of a 3-V dc component, a fundamental 2 sin t, and a second harmonic 1.5 sin 2t.
Draw accurately to scale one cycle of the waveform.
2.9. A spectrum component of a waveform is given by 3 sin(t20˚). Express this as the sum of sine and cosine terms.
2.10. A spectrum component of a waveform is given by 5 cos(t70˚). Express this as the sum of sine and cosine terms.
2.11. The nth harmonic of a waveform can be written as 7 cos(n0t35˚). Determine the corresponding anand bnamplitudes.
f(x)sin x for x .
TV video Music Speech
15 300 3400 20,000 4 × 106
106 105
104 103
Frequency, Hz (log scale) 102
10
1 107
Figure 2.18.1 Bandwidth requirements for some baseband signals.
2.12. The Fourier coefficients for the nth harmonic of a waveform are an35 V,bn45 V. Express the harmonic in cosine form.
2.13. For a waveform series An1/n2,a00 and n0. Given that all harmonics fromn 1 to 10 are present in the waveform, plot this using Eq. (2.4.6).
2.14. The Fourier series for a square wave of 1-V amplitude is
By comparing this with the series given in Eq. (2.7.1), sketch the square wave relative to the wave- form shown in Fig. 2.7.1(a).
2.15. The Fourier series for a half-wave rectified sinusoidal current wave of 1-A amplitude is
Draw accurately to scale the spectrum up to the eighth harmonic. What is the value of the dc component?
2.16. Write out the next three terms for the wave in Problem 2.15. Using Mathcad or any other suitable computer program (or writing your own), construct accurately to scale two cycles of the waveform, using harmonics up to and including the eleventh.
2.17. Calculate the rms value of the thirteenth harmonic of a square wave for which the peak-to-peak voltage is 10 V.
2.18. Calculate the rms value of the tenth harmonic of a sawtooth wave for which the peak-to-peak voltage is 10 V.
2.19. Interpret Eqs. (2.5.1), (2.5.2), and (2.5.3) for the waveform shown in Fig. 2.7.2(a) to determine which coefficients exist. Do not attempt to solve the integrals.
2.20. Repeat Problem 2.19, but with the square wave set up as an odd function.
2.21. Repeat Problem 2.19, for a 3-V peak, half-wave rectified sine wave. (Hints: Set up the waveform as an even function; use the expressions given in Problem 2.15 to find the dc component.)
2.22. Plot the envelope of the amplitude function given by Eq. (2.9.3) for a periodic time of 1 ms, a pulse width of 0.1 ms,V1Vand n1 to 30 inclusive.
2.23. Write down the defining equations for the sine function and for the sampling function. Evaluate (a) sine 0.3, (b) sine 0, (c) sine 1, (d) Sa(0.3), (e) Sa(0), (f) Sa(1).
2.24. Given that sine 0.2, determine the value of (a) and (b) Sa().
2.25. Plot the function sine xfor xin steps of 0.1 over the range 2.2 x 2.2.
2.26. One cycle of a periodic waveform is described by
f(x) cos x, for 0 x , and cosx, for x0 By applying rule 1 of Section 2.10, determine if the spectrum contains a dc component.
2.27. For the waveform of Problem 2.26, apply rules 2 and 3 of Section 2.10 to determine if the trigono- metric spectrum consists of sine or cosine terms.
2.28. Apply rules 2 and 3 of Section 2.10 to the waveform of Fig. 2.8.1 to determine if it contains only even or odd harmonics, or both.
i(t) 1 1
2sin 0t 2
cos 2130t cos 4350t cos 6570t …v(t) 4
sin 0t sin 330t sin 550t…2.29. Apply rule 4 of Section 2.10 to determine the extent of the harmonic content of the waveforms described by Eq. (2.8.1). Assume harmonics less than 1% of V can be ignored.
2.30. Apply rules 1 through 3 of Section 2.10 to the waveform of Problem 2.4 and draw the necessary conclusions.
2.31. Apply rules 1 through 3 of Section 2.10 to the waveform of Problem 2.5 and draw the necessary conclusions.
2.32. Determine the exponential Fourier coefficients for the sine wave of Problem 2.1.
2.33. Determine the exponential Fourier coefficients for the waveform of Problem 2.6.
2.34. Write out in exponential form the equations for the waveforms in Problems 2.8, 2.9, and 2.10.
2.35. Write out in exponential form the series given by Eqs. (2.7.1), (2.7.2), (2.8.1), and (2.9.1).
2.36. The amplitude of a harmonic term in an exponential Fourier series is cn3 V. Determine (a) the amplitude and (b) the rms value of the corresponding sinusoidal function.
2.37. The amplitude of a term in an exponential Fourier series is cn3 j4 V. Write out the equation for the corresponding trigonometric function.
2.38. The first two terms in an exponential Fourier series are co1 V and c12 j5 V. Write out the corresponding trigonometric terms.
2.39. If the fundamental frequency in the waveform of Problem 2.6 is 1000 Hz, at what rate should it be sampled in order to compute the spectrum from the samples?
2.40. Assuming that harmonics less than 10% of the ac peak amplitude V can be ignored, determine the number of samples to be taken for the square wave shown in Fig. 2.7.2(a) in order that the waveform may be analyzed through an fft routine. The periodic time of the waveform is 3 ms.
2.41. Repeat Problem 2.40 for the sawtooth waveform of Fig. 2.8.1(a).
2.42. The highest-frequency component in a speech signal spectrum is 4 kHz. At what rate should this sig- nal be sampled in order to compute the spectrum from the samples?
2.43. By using an fft computer routine, determine the spectrum for the waveform of Fig. 2.7.2(a). Compare the results with those obtained in Problem 2.19.
2.44. Repeat Problem 2.43 with the square wave set up as an odd function.
2.45. By using an fft computer routine, determine the spectrum for a 60-Hz, 3-V peak, half-wave rectified sine wave. Compare the results with those obtained in Problem 2.21.
2.46. A cosine wave has a periodic time of 1 s and a maximum value of 1 V. When this is passed through an amplifier, horizontal peak clipping is observed. Using the center of a positive peak as the time zero reference, the clipping extends to 0.1 s and starts again at 0.45 s, repeating in this fashion for every cycle. Sketch one cycle of the clipped waveform, and use a computer fft routine to find its spectrum.
2.47. Use Eqs. (2.5.1), (2.5.2), and (2.5.3) to find the theoretical spectrum for the waveform in Problem 2.46.
Compare the results in volts with those obtained in Problem 2.46 for the first 6 terms.
2.48. Explain what is meant by an energy signal. The voltage spectral density curve for a pulse can be approximated by a linear rise from 0 to 5 V/Hz over the frequency range from 0 to 5000 Hz, followed immediately by an exponential fall-off expressed as 5 exp(1 f/5000) V/Hz for f5000 Hz. Sketch the voltage spectral density curve and the energy spectral density curve, assuming the voltage is developed across a load of 1 .
2.49. Determine the energy in the pulse described in Problem 2.48.
2.50. Using Eq. (2.13.3), plot the voltage spectral density for A1 V, 1 ms over the frequency range 2.2/ f 2.2/in steps of 0.1/. State clearly the units on the graph axes.
2.51. A rectangular pulse has an amplitude of 5 V and a width of 3 ms. Determine the voltage spectral density at frequencies of (a) 30 Hz, (b) 100 Hz, and (c) 3000 Hz, stating clearly the units used.
2.52. Determine the spectrum for a 3-V, 0.5-s rectangular pulse such as shown in Fig. 2.13.1(a), using a computer fft routine. Compare the results with the theoretical spectrum given by Eq. (2.13.3).
2.53. Explain what is meant by a power signal. Give one example each of a deterministic and a nondeter- ministic power signal.
2.54. The power spectral density curve for a nondeterministic signal can be approximated by a linear rise from 0 to 3 J over the frequency range from 0 to 300 Hz, remaining constant at 3 J up to 3000 Hz, followed by a linear drop-off reaching 0 J again at 8000 Hz. Sketch the power spectral density curve and determine the average signal power.
2.55. A waveform consists of an infinitely long, random sequence of pulses, the pulses being independent of one another and the waveform having zero dc level. The basic pulse has a voltage spectrum den- sity the magnitude of which is rectangular in shape, being 2 mV/Hz from 100 to 3000 Hz and zero outside these limits. Given that the pulse width in the time domain is 6 ms, determine the average power in the waveform.
2.56. Can the pulse shape in the time domain be determined from the information given in Problem 2.55?
2.57. Using Eq. (2.17.2), plot the power spectrum density curve for a binary waveform that meets the conditions stated in the text, over the frequency range 0.1 Hz to 10 kHz. The basic rectangular pulse has an amplitude of 5 volts and a duration of 3 milliseconds.
2.58. Briefly describe the bandwidth requirements for speech and video signals. How many speech chan- nels, approximately, could be fitted into the bandwidth required for a TV video signal?
2.59. Plot the complex sine wave v(t) using MATLAB, for m5 and m =10.
Letan and fo100Hz.
2.60. The MATLAB function fft(.) can be used to obtain the Fourier transform of a discrete sequence.
Applyfft(.)on the discrete sequence v(n) obtained from Problem 2.1. (Hint: Let t[0:0.01:5]) 2.61. Consider a sinusoidal signal,e(t) 10sin(2000t). Plot the entire waveform and one full wave of e(t).
2.62. Assume that an arbitrary waveform sequence is available as one row vector,v(n). Obtain and plot 10 replicasof v(n). (Hint: Use the column operator (:) in MATLAB. Vx V * ones(1, 10); VxVx(:);
stem(Vx);)
2.63. Generate one wave of the arbitrary waveform shown in Figure P2.63, using MATLAB.
1 n2
mn1an cos (2nfot)
0 1 2 3 4 5
-1 +1
Figure P2.63
2.64. Replicate the waveform given above 5 times and plot it.
2.65. Using MATLAB, clip the lower half cycle of a sine wave and plot it.
2.66. One cycle of a periodic waveform is described by v(x) sin(x),for0 x and sin(x),for x 2. Plotv(x) using MATLAB. Check whether the waveform has a dc value.
2.67. Plot, using MATLAB, a Gaussian pulse train consisting of 10 pulses. (Hint: Use gauspuls(.) function along with the solution to Problem 2.4.)
2.68. Obtain the dc value of the waveform depicted in Problem 2.7. Verify the result using MATLAB.
(Hint: Use mean(.) function.)