Fourier Transform of radio pulses

[fourierpwn]

Hi all, I am currently bed ridden with the flu and have missed a class this week for my communication systems class. The solutions to these questions are only given in class as they're worked through on the board.

I'm a little stuck with this question (my name seems to be fitting quite well right now; pwned by Fourier  ).

Could anyone please offer some insight on how to approach this question?



Edit: I think what's throwing me off the most is the Pi of t as the amplitude of the signals. I've never seen that before, other than Pi(t/T) which is a rect function. I feel like perhaps it means it's a rect pulse of sine/cosine waves?

My approach so far has been to ignore the pi(t)/amplitude and use Euler's formula to rewrite the cos/sin(2pif0t). Which results in a shifted sinc function?

[tec5c]

Edit: I think what's throwing me off the most is the Pi of t as the amplitude of the signals. I've never seen that before, other than Pi(t/T) which is a rect function. I feel like perhaps it means it's a rect pulse of sine/cosine waves?

My approach so far has been to ignore the pi(t)/amplitude and use Euler's formula to rewrite the cos/sin(2pif0t). Which results in a shifted sinc function?

You're on the right track.

So you use Euler's formula for each pulse, then take the fourier transform of that function (giving sum and difference...) and you're essentially done.

While there are signs of the 'frequency shift property', it does not result in a sinc function (I don't think? Sorry, it's late.)

Make sure you don't forget that when using Euler's formula that sin(t) results in 1/2j [exp(j*omega*t) MINUS exp(-j*omega*t)] as it is a common mistake to forget the 2j and minus sign.

Hope this helps.

[KD0CAC John]

Not that I know the subject , but found this site to help learn .

http://www.fourier-series.com/

[unitedatoms]

The inverted letter "U" may not be Pi, since in the same text actual Pi is drawn normally ...2Pif...
https://en.wikipedia.org/wiki/Rectangular_function

I'd think the same as original poster, that inverted "U" stands for pulse. So, yes the result should be Sinc enveloped series of harmonics for first one and cosc (cosine cardinal enveloped harmonics) for second one.

The answers are
http://www.wolframalpha.com/input/?i=fourier+exp+transforms+of+Pi%28x%29+sin%282+pi+f+t%29

http://www.wolframalpha.com/input/?i=fourier%20exp%20transforms%20of%20Pi%28x%29%20cos%282%20pi%20f%20t%29&lk=2

Or put "fourier transformation of rectangular function * sin(2*Pi*f*t)" into search window in WolframAlpha

And "fourier transformation of rectangular function * cos(2*Pi*f*t)"

[jpb]

In standard maths the capital pi means a product over terms - similar to the use of a capital sigma to indicate a summation.

If you take the log of such a product you get a summation of logs. Taking a Fourier transform might be a bit tricky - I seem to remember that the Fourier transform of a product is a convolution (a sliding of one function across another) but I don't have any of my maths books with me at present.

Given that the terms are all sines (or cosines) you might find that most terms cancel in the transform.

[DJohn]

In standard maths the capital pi means a product over terms - similar to the use of a capital sigma to indicate a summation.

Not when it's used as a function.  In this context, it's almost certainly the rectangle function Pi(x) = 0 if x < -1/2 or x > 1/2, Pi(x) = 1 if -1/2 < x < 1/2, Pi(x) = 1/2 if x = +/- 1/2.  It's being used here as a windowing function, to restrict an infinite signal to a finite region of time.

The Fourier transform of a function is the integral from -infinity to +infinity.  Since Pi(x) = 0 outside the interval [-1/2, 1/2] (and 1 inside it), you can replace the limits with 1/2 to 1/2.  Form there you're on your own - I'm allergic to integrals.