 Sampling and quantization Seminary 2. Problem 2.1 Typical errors in reconstruction: Leaking and aliasing We have a transmission system with f s =8 kHz.

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Sampling and quantization Seminary 2

Problem 2.1 Typical errors in reconstruction: Leaking and aliasing We have a transmission system with f s =8 kHz sampling frequency. The system’s input (anti-aliasing) and output (reconstruction) filter has the same characteristics: a) Give the all spectral components and their amplitudes at the output if the input signal is a 1 kHz sinusoidal signal with 2V amplitude. b) What will be the output if we grow the input signal’s frequency to 4.5 kHz?

Problem 2.2 How to choose the sampling frequency The spectral density of a stationary voltage signal is zero on all positive frequencies, except in the band between 0 and 3 kHz and 7 to 8 kHz where is constant s 0. a) Determine the spectral density on the negative frequencies. b) Give the value of s 0 if the signal’s power is 0.2 mW and the load is 50 Ω. c) Give all the possible sampling frequency, provided the sampled signal should be perfectly reconstructed. Give the characteristics of the suitable reconstruction filter as well.

Problem 2.3 Sampling of non-baseband signal We have to digitalize a signal having spectral components between 19 and 25 kHz only. a) What is the lowest possible sampling frequency allowing the perfect reconstruction? Give the characteristics of the reconstruction filter. b) What are the possible sampling frequencies? c) Applying same quantizer and reconstruction filter, compare the quantization noise when the minimal and when 50 kHz sampling is applied. d) At 50 kHz sampling frequency and applying a B=6 kHz (ideal) band pass reconstruction filter, how many bits we need in a linear quantizer to reach an SNR ≥ 90dB if the signal’s crest factor is 5?

Problem 2.4 Oversampling We digitalize a baseband signal of 10 kHz bandwidth applying a 12 bit linear quantizer. The signal’s crest factor is 5. a) What is the signal to quantization noise ratio of the reconstructed signal if we use the minimum sampling frequency? b) What sampling frequency should be applied to reach a 69 dB SNR at least? c) How many bit quantizer should be used with the minimum sampling frequency to reach 69 dB SNR?

Shaping the quantization noise: Sigma-Delta Converter If H(z) is low-pass, then 1 – H(z) is high-pass

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