Impulse response from frequency response

In summary, the question is asking for help in deriving the inverse Fourier transform of a sinc function and an exponential function to obtain the impulse response from a given frequency response and to calculate the output for a given input. The approach for solving this problem involves using Laplace transform and convolution, but the work shown may not be correct. In addition, the method used for finding the system output in part (b) is invalid as it mixes the frequency and time domain. The correct approach is to use convolution for combining the input and impulse response in the time domain, or multiplication in the frequency domain.
  • #1
aguntuk
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0

Homework Statement



I am having some problems to derive Inverse Fourier transform of sinc function & exponential function. It's actually for getting the Impulse response from the given frequency response [which comprises of both sinc function & exponential function]. Also need to know the output y(t) for the given input. Can anyone help me out?

The question is in the attachment.

Homework Equations



(a) The impulse response is the inverse transform [IFT] of the transfer function. I think IFT will lead me to get the impulse response.

(b)

The Attempt at a Solution



Solution:

(a)

I stepped forwards as following:

h_0 (t)= F^(-1) [H(jω)]
= 1/2π ∫_(-∞)^(+∞)▒〖T_1 . e^(-jω T_1/2) . sin⁡〖ω T_1/2〗/(ω T_1/2)〗 .〖 e〗^(-jωt) dω
= T_1/2π ∫_(-∞)^(+∞)▒〖e^(jω(t-T_1/2)) dω .∫_(-∞)^(+∞)▒sin⁡〖ω T_1/2〗/(ω T_1/2)〗 dω
= T_1/2π δ(t-T_1/2) ∫_(-∞)^(+∞)▒sin⁡〖ω T_1/2〗/(ω T_1/2) dω

I could not get the sinc function [sin⁡〖ω T_1/2〗/(ω T_1/2)] to Fourier transform and also the derivation from time lapse exponential function [e^(jω(t-T_1/2))] to delta function [δ(t-T_1/2)]. Can anyone help me out the full derivatives of the problem?(b) Is it the right equation to get the system output y (t) as follows?

y(t)= r(t)*[H(jω)]
= A T_1 . e^(-jω T_1/2) . sin⁡〖ω T_1/2〗/(ω T_1/2)
= A T_1 . e^(-jω T_1/2) . sinc(ω T_1/2)

Is the approach ok? And how can it be more simplified?
 

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  • #2
I'm having a little bit of trouble following your work, so I am going to make some quick comments.

for part a. typically it is easier to use Laplace (at least for me). If you don't know laplace then i guess your method is ok. note: i did not check your work for part a. the method is correct. the work may not be.

for part b. You are mixing the frequency and time domain, so your method is invalid.

y(jw)=r(jw).H(jw)
or
y(t)=r(t)*H(t)

H(t)= impulse response of system.
if you are working in the time domain you use convolution to combine the r(t) and H(t) signal. if you are working in the frequency domain you can simply use multiplication.

not r(t)=A*(u(t)-u(t-T))
 

FAQ: Impulse response from frequency response

1. What is impulse response and frequency response?

Impulse response is the output of a system when an impulse (a signal with extremely short duration) is applied as its input. It represents how the system will react to any input signal. Frequency response, on the other hand, is the measure of how a system responds to different frequencies of input signals. It shows the amplitude and phase of the output signal as a function of frequency.

2. How are impulse response and frequency response related?

Impulse response and frequency response are closely related. The frequency response can be obtained by taking the Fourier transform of the impulse response. In other words, the frequency response is the spectrum of the impulse response. This means that the frequency response contains all the information about the system's behavior in the frequency domain.

3. How is impulse response from frequency response calculated?

The impulse response can be obtained from the frequency response using the inverse Fourier transform. This process is also known as deconvolution. By taking the inverse Fourier transform of the frequency response, we can retrieve the impulse response of the system.

4. What are the applications of impulse response and frequency response?

Impulse response and frequency response are widely used in the field of signal processing and system analysis. They are used to characterize and understand the behavior of various systems such as filters, amplifiers, and communication channels. They are also used in the design and testing of audio and acoustic systems.

5. How does the shape of the frequency response affect the impulse response?

The shape of the frequency response can give us insights into the behavior of the impulse response. For example, a system with a flat frequency response (constant gain across all frequencies) will have an impulse response with a single peak, indicating that it will amplify all frequencies equally. On the other hand, a system with a high-pass filter frequency response will have an impulse response with a sharp peak, indicating that it will amplify high frequencies more than low frequencies.

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