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Demodulated signal A B Sampler Threshold decision circuit
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Detector output
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shown in block schematic form in Fig. 10.21 (Franks, 1980). The filters A and B form part of the normal signal filtering (e.g., raised-cosine filtering). The signal for the bit timing recovery is tapped from the junction between A and B and passed along a separate branch which consists of a filter, a squaring circuit, and a bandpass filter which is sharply tuned to the clock frequency component present in the spectrum of the squared signal. This is then used to synchronize the clocking circuit, the output of which clocks the sampler in the detector branch. The early-late gate circuit provides a method of recovering bit timing which does not rely on a clocking component in the spectrum of the received waveform. The circuit utilizes a feedback loop in which the magnitude changes in the outputs from matched filters control the frequency of a local clocking circuit (for an elementary description see, for example, Roddy and Coolen, 1995). Detailed analyses of these and other methods will be found in Franks (1980) and Gagliardi (1991). 10.9 Problems and Exercises
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10.1. For a test pattern consisting of alternating binary 1s and 0s, determine the frequency spectra in terms of the bit period Tb for the following signal formats: (a) unipolar; (b) polar NRZ; (c) polar RZ; (d) Manchester. 10.2. Plot the raised-cosine frequency response Eq. (10.9), for a bit rate of 1 b/s and a roll of factor of 1, for a symbol rate equal to the bit rate. Use the inverse Fourier transform to determine the shape of the pulse time waveform. 10.3. Plot the compressor transfer characteristics for The -law compression characteristic is given by
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v0 sign(vi) ln(1 ln(1 Zvi Z) )
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Ten
where v0 is the output voltage normalized to the maximum output voltage, and vi is the input voltage normalized to the maximum input voltage. The A-law characteristic is given by
A Zvi Z 1 ln A 1 A
sign(vi)
for 0
Zvi Z
1 1 ln (A Zvi Z) ln A 1 A
sign(vi)
Zvi Z
10.4. Write down the expander transfer characteristics corresponding to the compressor characteristics given in Prob. 10.3. 10.5. Assuming that the normalized levels shown in Fig. 10.4b represent millivolts, write out the digitally encoded words for input levels of (a) 90 mV; (b) 100 mV; (c) 190 mV; (d) 3000 mV. 10.6. Determine the decoded output voltage levels for the input levels given in Prob. 10.5. Determine also the quantization error in each case. 10.7. (a) A test tone having the full peak-to-peak range is applied to a PCM system. If the number of bits per sample is 8, determine the quantization S/N. Assume uniform sampling of step size V, for which the mean square noise voltage is ( V)2/12. (b) Given that a raised-cosine filter is used with 1, determine the bandwidth expansion factor B/W, where B is the PCM bandwidth and W is the upper cutoff frequency of the input. 10.8. A PCM signal uses the polar NRZ format. Following optimal filtering, the [Eb/N0] at the input to the receiver decision detector is 10 dB. Determine the bit error rate (BER) at the output of the decision detector. 10.9. Using Eq. (10.18), calculate the probability of bit error for [Eb/N0] values of (a) 0 dB, (b) 10 dB, and (c) 40 dB. 10.10. A PCM system uses 8 bits per sample and polar NRZ transmission. Determine the output [S/N] for [Eb/N0] values of (a) 0 dB, (b) 10 dB, and (c) 40 dB at the input to the decision detector. 10.11. A binary periodic waveform of period 3Tb is low-pass filtered before being applied to a BPSK modulator. The low-pass filter cuts off at B 0.5/Tb. Derive the trigonometric expansion for the modulated wave, showing that only side frequencies and no carrier are present. Given that the bit period is 100 ms and the carrier frequency is 100 kHz, sketch the spectrum, showing the frequencies to scale.
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