Assignment 1 - Dsp

Assignment 1 1. Given an analog signal: x(t) = 10Sin2πt + 10Sin8πt + 5Sin12πt

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Sampling x(t) at the frequency rate fs = 5Hz. Find the output signal xa(t) so that xa(t) is an alias of x(t), and prove that when sampling, two signal x(t) and xa(t) will give the same sampled signal. Repeat with fs = 10Hz. 2. The signal x(t) = Sin(6πt)[1 + 2Cos(4πt)] , t[ms] is sampled at the sampling rate f s = 4KHz. The sampled signal is passed through the ideal re-constructor. Find the output signal of the re-constructor. 3. Consider the sound wave: x(t) = Sin10πt + Sin30πt + Sin50πt + Sin80πt

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a. If x(t) is sampled at the frequency rate fs = 80Hz, does this frequency satisfy the sampling theorem? Find the discrete-time signal. b. If x(t) is sampled at the frequency rate fs = 40Hz. Find the analog signal after reconstructing. Assume that the reconstruction process is ideal. 4. Consider the following sound wave, where t is in milliseconds. x(t) = Sin(16πt) + Sin(30πt) + Sin(60πt). This is pre-filtered by an analog anti-aliasing pre-filter H(f) and then sampled at an audio rate of 44KHz. The resulting samples are immediately reconstructed using an ideal re-constructor. Determine the output ya(t) of the re-constructor in the following cases and compare it with the original x(t). a. When there is no pre-filter (H(f) = 1). b. When H(f) is an ideal pre-filter with cut off of 20KHz. c. When H(f) is a practical pre-filter that has a flat pass-band up to 20KHz and attenuates at a rate of 42dB/octave beyond 20KHz. Ignore the effects of the phase response of the filter. 5. It is desired to pick an ADC for a DSP application that meets the following specifications: The full-scale range of the converter should be 10 volts and the Root mean square (RMS) quantization error should be kept below 1 milli-volt. Determine: a. The number of bits should the converter have? b. The actual RMS error of the A/D converter? c. The dynamic range in dB of the converter?