Question

(b) Perform convolution to obtain the discreet if input x[n] = [1 3 2 1] and impulse response, h[n] signal output of y[n], [1 -4 2]. [3 marks] (c) An analogue signal is sampled every 50ms for a duration of 1000 seconds. i) Calculate how much data (samples) are collected. [2 marks] If a Discrete Fourier Transform (DFT) is performed what is the maximum frequency information that can be obtained? [2 marks] Calculate the minimum frequency (the frequency resolution) a DFT will directly obtain information about. [3 marks]

Answer 1

b). Given

The formula for convolution is given as :

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applying this formula we obtail

$y[0]=\sum_{-\infty}^{\infty}x[k]h[-k]= 1$

y[1] x[k]h[1 - k] -1

$y[2]=\sum_{-\infty}^{\infty}x[k]h[2-k]= -8$

$y[3]=\sum_{-\infty}^{\infty}x[k]h[3-k]= -1$

$y[4]=\sum_{-\infty}^{\infty}x[k]h[4-k]= 0$

$y[5]=\sum_{-\infty}^{\infty}x[k]h[5-k]= 2$

c).

i). Number of samples = $\frac{\text{total signal duration}}{\text{sampling duration}} = \frac{1000}{50\times 10^{-3}}= 20000$

ii). Highest Frequency is the sampling frequency which is given by inverse of sampling duration i.e,

$\text{Sampling rate}= \frac{1}{\text{Sampling duration}}=\frac{1}{50\times 10^{-3}} =20 \ Hz$

iii).

. $\text{frequency resolution}= \frac{\text{Sampling rate}}{\text{Number of samples}}=\frac{1}{50\times 10^{-3}\times 20000} \\=1 \times 10^{-3 } Hz$