Question

This is in electrical engineering signals, I NEED IT ASAP PLEASE!!

(b) A continuous-time LTT system has an input (t) and an output y(t). The frequency response of the system is (w). 0) (1 point) Write an equation that describes the relationship between the Fourier trans- forms of the input and output (X (jw) and Y Gw)) and the frequency response (jw). For the rest of the problem, assume that X(jw) and (w) are as shown in the plots below. X(w) (jw) 4 3 3 1 -2x (100) 2(30) 0 +2(150) +20 (100) 2 (100) - 2 (50) 0 + 2 (50) 2 (100) (ii) (4 points) Determine and sketch the Fourier transform Y Gw) of the system output. (iii) (5 points) Suppose that instead of processing a(t) with the contimous-time system (as in the pre- vious part), you want to achieve the same results by processing the signal with a sampled system. In other words, you process it using the sampling, filtering, and reconstruction system shown below W filter Curresite to impube M CT ile -- --T) C/D converter D/C converter Assume that the sampling frequency is fs - 400 Hz and the DT filter has the frequency response 17(e)shown below. (en) 1 12 - -2 +3 +it+*+2 + Will the Fourier transform of the output of the sampledd system (Y.Gw)) match the Fourier transform Y Gw) you computed in part b-ii above? If so, justify your answer. If not, say how the sampled system could be altered so that ye(jw) = Y(w).

Answer 1

1) Given input signal is x(t), and output y(t). The frequency response of the system is H(jomega).

If H(jomega) has impulse response of h(t), the relation between the input and output signals is given by using the convolution in time domain as:

Convolution in time domain will correspond to multiplication in frequency domain, Hence, the relation in frequency domain is:

The above gives the relation of fourier transform of input and output signals.

2) given X(jomega)

and H(jomega)

By using the relation:

The fourier transform of output signal is the multiplication of fourier transform of the input and the system transfer function:

H(jomega) is zero in range 2pi100 and 2pi50, in that range the output fourier transform is zero.

Hence Y(jomega) =

c) Note that any sampled signal can be reconstructed only if it satisfies sampling theorem.

The sampling theroem states that the sampling frequency should be greater than or equal to twice the message frequency or message signal band width.

Fb is the bandwidth of message signal

From the fourier transform of the input signal X(jomega), observe that the band width of the input signal is

200, since w = 2*pi*f, from X(jomega) f lies between +100 and -100.

Hence the sampling frequency should be,

Given Fs = 400,

Given sampling frequency is 400, which satisfies the sampling theorem ( signal has minimum sampling frequency).

To match the output signal the filter band width also should be reduced by same amount. The band width of given digital filter is

pi/2+pi/2 = pi

But the analog filter has bandwidth of

100pi+100pi = 200pi

The digital filter bandwidth is

To acheive this band width the cutoff frequency of filter should be

pi/4, hence

pi/4+pi/4 = pi/2

By changing the filter, cutoff frequency, we can reconstruct the output signal.