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EE 3008A Linear Systems and Signal Processing

Department of Electrical Engineering

Due date: Thursday Nov. 9, 2023

Homework 4

1. For a given signal f(t) = cosw1 t + cos2w1 t, determine the minimum sampling frequency for subsequent perfect signal reconstruction.

2. A signal f(t) = 10cos20pt cos200pt is sampled with sampling frequency 450Hz.

(a) If an ideal low-pass filter is used to recover f(t) from the sampled signal, determine the bandwidth of the low-pass filter required;

(b) What is the Nyquist sampling rate for f(t)?

3. Current technologies can support sampling circuits with sampling speed up to 160 G Samples/s. What is the maximum signal bandwidth that can be sampled, transmitted through a communication channel, received and recovered correctly?

4. A signal x(t) with bandwidth 10 rad/s is sampled at a rate of 40 samples/s. The sampled signal xp (t) is transmitted and received perfectly and ideal interpolation is used to reconstruct x(t) . Show that 

5. A message signal x(t) is sampled to give xp (t) . To reconstruct the original signal, xp (t) then passes through an interpolation filter with impulse response h(t) to give xr (t) as shown in the figure below.

(a) Show that xr (t) is a piecewise linear waveform which is formed by connecting neighboring sample values through straight lines.

(b) What is the transfer function of the interpolation filter H(ω) ?

(c) xr (t) is not the same as the original signal x(t) but an equalizing filter Heq (ω) can be used to recover x(t) . What is Heq (ω) ?

Part of this problem can be related to the results of Homework 1 Q11b and Q12.

6. A TDM system consists of 24 transmission channels and a sampling rate of 8kHz is used. The bandwidth of the signal for each channel is below 3.3kHz. Determine the minimum bit rate required to transmit this TDM signal if we use a) 16 quantization levels b) 128 quantization levels for each sample.

7. Five signals are combined to be transmitted in a TDM system, the combined signals will pass through a low-pass filter. Three channels are used to transmit the signals of frequency range between 300 to 3300 Hz and the other two channels transmit the signals of 50 Hz to 10 kHz range.

(a) What is the minimum sampling rate required?

(b) What is the minimum bandwidth of the low-pass filter required corresponding to the minimum sampling rate?

8. The information in an analog voltage waveform is to be transmitted over a PCM system with a ±0.1% accuracy. The analog waveform has a bandwidth of 100 Hz and an amplitude range of –10 to 10 V.

(a) Determine the minimum sampling rate needed;

(b) Determine the number of bits needed to represent each sample;

(c) Determine the minimum bit rate required in PCM signal;

9. For a signal f(t) = 9 + Amcoswmt, with Am £ 10, to be sampled and quantized into 41 binary levels or more.

(a) Determine the number of bits needed in each sample;

(b) What are the values of the highest and lowest quantized levels Vmax and Vmin if the quantized boundaries are centered to [f(t)max + f(t)min] / 2;