• Document: ELE745 Assignment and Lab Manual
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ELE745 Assignment and Lab Manual August 22, 2010 CONTENTS 1. Assignment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Assignment 1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Assignment 1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Assignment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Assignment 2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Assignment 2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3. Assignment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1 Assignment 3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Assignment 3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4. Assignment 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Assignment 4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Assignment 4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5. Experiment 1: Sampling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6. Experiment 2: Binary Signalling Formats . . . . . . . . . . . . . . . . . . . . . . . . . 41 7. Experiment 3: Matched Filter and Bit Error Rate (BER) . . . . . . . . . . . . . . . . 49 8. Experiment 4: Digital Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1. ASSIGNMENT 1 1.1 Assignment 1 Problems 1. Lathi, 6.1-1 Following figure shows Fourier spectra of signals g1 (t) and g2 (t). Determine the Nyquist interval and the sampling rate for signals g1 (t), g2 (t), g12 (t), g23 (t), and g1 (t)g2 (t). 2. Lathi, 6.1-2 Determine the Nyquist sampling rate and the Nyquist sampling interval for the signals: (a) sinc(100πt); (b) sinc2 (100πt); (c) sinc(100πt) + sinc(50πt); (d) sinc(100πt) + 3sinc2 (60πt); (e) sinc(50πt) sinc(100πt). 3. Lathi, 6.1-4 A signal g(t) = sinc2 (5πt) is sampled (using uniformly spaced impulses) at a rate of: (i) 5 Hz; (ii) 10 Hz; (iii) 20 Hz. For each of the three case: (a) Sketch the sampled signal; (b) Sketch the spectrum of the sampled signal; (c) Explain whether you can recover the signal g(t) from the sampled signal; 1. Assignment 1 2 (d) If the sampled signal is passed through an ideal low-pass filter of bandwidth 5 Hz, sketch the spectrum of the output signal. 4. Lathi, 6.1-8 Prove that a signal cannot be simultaneously time-limited and band-limited. Hint: show that contrary assumption leads to contradiction. Assume a signal simultaneously time- limited and band-limited so that G(ω) = 0 for |ω| > 2πB. In this case G(ω) = G(ω)rect(ω/4πB 0 ) for B 0 > B. This means that g(t) is equal to g(t) ∗ 2B 0 sinc(2πB 0 t). Show that the latter cannot be time-limited. 5. Lathi, 6.2-2 A compact disc (CD) records audio signal digitally by using PCM. Assume the audio signal bandwidth to be 15 kHz. (a) What is the Nyquist rate? (b) If the Nyquist samples are quantized into L = 65, 536 levels and then binary coded, determine the number of binary digits required to encode a sample. (c) Determine the number of binary digits per second (bit/s) required to encode the audio signal. (d) For practical reasons, signals are sampled at a rate well above the Nyquist rate. Practical CDs use 44,100 samples per second. If L = 65, 536, determine the number of bits per second required to encode the signal. 6. Lathi, 6.2-3 A television signal (videl and audio) has a bandwidth of 4.5 MHz. This signal is sampled, quantized, and binary coded to obtain a PCM signal. (a) What is the sampling rate if the signal is to be sampled at a rate 20% above the Nyquist rate. (b) If the samples are quantized into 1024 levels, determine the number of binary pulses required to encode each sample. (c) Determine the binary pulse rate (bits per second) of the binary-coded signal, and the minimum bandwidth required to transmit this signal. 7. Lathi, 6.2-4 Five telemetry signal, each of bandwidth 1 kHz, are to be transmitted simultaneously by binary PCM. The maximum tolerable error in sample amplitudes is 0.2% of the peak signal amplitude. The signals must be sampled at least 20% above the Nyquist rate. 1. Assignment 1 3 Framing and synchronizing requires an additional 0.5% extra bits. Determine the min- imum possible data rate (bits per second) that must be transmitted, and the minimum bandwidth required to transmit this signal.

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