Fourier Transform Analysis of CT Systems

FOURIES TRANSFORM ANALYSIS OF CT SYSTEMS

Fourier transform X(ω) of signal x(t) gives the frequency spectrum of the signal.

i. Output of continuous time system is represented as

y(t) = x(t) * h(t)

ii. By using convolution theorem, the above equation becomes,

Y(ω) = X(ω).H(ω)

By taking inverse Fourier transform of y(ω) we can obtain the output y(t) of the system.

Frequency Response

H(ω) is the frequency response of LTI - CT system. It is also known as system

transfer function.

We know that, y(ω) = X(ω).H(ω)

H(ω) = Y(ω)/X(ω)

Here Y(ω) ⇒ Fourier transform of output y(t)

X(ω) ⇒ Fourier transform of input x(t)

H(ω) ⇒ Fourier transform of h(t) (i.e) impulse response.

Solution of Differential Equations

Consider the differential equation

Apply differentiation in time domain property of Fourier transform to above equation.

Problems Based on Fourier Transform Analysis of CT Systems

Problem 1:

Determine frequency response and impulse response for the system described by the following differential equation. Assume zero initial condition.

Solution:

(i) Frequency response - H(ω)

Taking Fourier transform of given differential equation.

(ii) Impulse Response [h(t)]

Inverse Fourier transform of H(ω) is h(t).

Taking Inverse Fourier transform (1)

Problem 2:

The system produces the output of y(t) = e^-t u(t) for an input of x(t) = e^-2t u(t). Determine the impulse response and frequency response of the system. [May 13-Marks 16]

Solution:

(i) Frequency Response - H(ω)

Taking Fourier transform of (1)

Taking Fourier transform of (2)

Frequency response H(ω) = Y(ω)/X(ω)

(ii) Magnitude of Frequency Response:

Problem 3:

Consider the continuous time LTI system described by 

Obtain an output for the input x(t) = e^-t u(t) using fourier transform.

Solution:

Taking Fourier Transform of (1)

Taking Fourier Transform

Now substitute X(ω) in Y(ω)

Taking inverse Fourier transform

Problem 4:

Consider the continuous time LTI system described by Using fourier transform, find the output y(t) to each of the following input signals.

(i) x(t) = e^-t u(t) (ii) x(t) = u(t)

Solution:

Taking Fourier Transform

(i) x(t) = e^-t u(t)

Taking Fourier Transform.

Now substitute X(ω) in (1)

Taking inverse fourier transform

(ii) x(t) = u(t)

Taking Fourier Transform

Now substitute X(ω) in (1)

Problem 5:

A stable LTI system is described by the differential equation.

Find the frequency response and impulse response using Fourier transform. [Dec 13-8Marks]

Solution:

Taking Fourier transform

Taking inverse Fourier transform.

Problem 6:

Frequency response of the causal LTI system is give by H(jω) = . The system produces an output of . Determine the input. [May-08-6 Marks]

Solution:

Taking Fourier transform

Taking inverse Fourier transform

Problem 7:

The impulse response of the continuous time system is given as . Determine the frequency response and plot magnitude and phase plot.

Solution:

Taking Fourier transform of impulse response.

Magnitude of frequency response