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Fourier Transform properties

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Linspire

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Guys,
I'm still confused to solve the following question :

Fourier transform properties 1.PNG
solution 1.PNG
solution 2.PNG


Regards
Linspire
 

Hii Linspire...Well, the questions that you have sent can be solved if you checked the fourier tables. Check this link http://www.ece.uah.edu/courses/ee426/fourier.pdf . First of one, identifies the case of the question with the proporty correspondent, then applie to it...It is really easy to do that...Any specifi question , you can ask me.

Greetings,
Renzo
 

Linspire said:
Guys,
I'm still confused to solve the following question :

View attachment 65049
View attachment 65050
View attachment 65051


Regards
Linspire
Click to expand...

What's your specific Problem ?

\[sinc(x)=\frac{sin(x)}{x}\]

and here is one sample PSD plot



---------- Post added at 14:31 ---------- Previous post was at 14:17 ----------

To find the Fourier transform ,some important properties of fourier transform are used like

1.If the \[ F(x(t))=X(f) then F(x(t{\pm}t_0))=X(f) \exp{j {\omega}{\pm} t_0}\]

Here is one good lecture note
h**p://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
 

Linspire said:
How to solve Question (ii) ?
Click to expand...

If Fourier transform of \[ Q(t) is Q(f) then F(Q(3t)) is \frac{1}{3}Q(f/3) \]

---------- Post added at 19:56 ---------- Previous post was at 19:47 ----------

And another interesting fact ,the gaussian is an example of self reciprocal function ,both the function and it's transform
have same form .
 

what's the properties you applied ?
 

I'm wondering how does "sinc^2 " function come as part of solution ?
I thought solution should be straight forward.
 

Sinc^2 is Generally the fourier transform of some triangular waveform
 

But my question didnt state it's triangular waveform ?
or is it alphabet Q(t) , actually standard representation for a triangular form ?

---------- Post added at 22:52 ---------- Previous post was at 22:43 ----------

Opps, sorry , I know already, in my notes state Qt is actually a unit height of triangular pulse.
 

How about question (iii) and (iv) ?

For part (iii), I dont get it why the solution gives (4 sinc (2W)) , is it because of P2(t) term ?
For part (iv), why the solution ignore partial fraction the 3rd term which Fourier transform table shown below
solution 3.PNG
 

Hi Linspire ,

What's Q(t) ? ,If the transform of Q(t) is Q(f) then the scale change theorem says that
transform of Q(3t) is 1/3 Q(f/3) ...

---------- Post added at 16:26 ---------- Previous post was at 16:00 ----------
Linspire said:
How about question (iii) and (iv) ?

For part (iii), I dont get it why the solution gives (4 sinc (2W)) , is it because of P2(t) term ?
For part (iv), why the solution ignore partial fraction the 3rd term which Fourier transform table shown below
View attachment 65071
Click to expand...

About third Question

\[ P_2(t)*Q_1(t)\]

Ans is
\[P_2(f).Q_1(f)\]

Where \[P_2(f),Q_1(f) \] denotes their individual Fourier transform .
ie convolution in time domain means multiplication in Frequency Domain.

---------- Post added at 16:36 ---------- Previous post was at 16:26 ----------

About 4th Question -Apply the modulation theorem

iv)\[Q_1(t)cos(40t)\]

Ans
\[ \frac{1}{2} Q_1(f-40)+\frac{1}{2}Q_1(f+40)\]

Where \[Q_1(f)\] Denotes the Fourier Transform of \[ Q_1(t)\]
 
Thank you blooz.
It's clear enough for me understand.
 

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