Re: finding Fourier Series.
- To: mathgroup at smc.vnet.net
- Subject: [mg63904] Re: finding Fourier Series.
- From: "Mariusz Jankowski" <mjankowski at usm.maine.edu>
- Date: Sat, 21 Jan 2006 01:50:54 -0500 (EST)
- Organization: University of Southern Maine
- References: <dqks93$ptf$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
To answer your second question first, in principle, you cannot evaluate the
FFT symbolically, because it is a numerical algorithm for calculating the
so-called discrete Fourier transform (DFT). When you ask for a symbolic FFT
you probably mean the so-called discrete-time Fourier transform (DTFT). To
get this simply use the definition, for example given your specific
sequence,
Sum[ a^n Exp[- I * w * n], {n, 0, Infinity}]
where w is the frequency in radians/second.
To evaluate the discrete-time Fourier series (DTFS) of a periodic sequence
of period N you may use the function Fourier (as DTFS and DFT are closely
related) to get a numerical result. But again to obtain a symbolic result
you should use the definition. For a given sequence x[n] the DTFS X[k] is
1/N Sum[x[n] Exp[- I*2*PI/N*n*k ], {n, 0, N-1}]
For a review of these topics go to
http://www.usm.maine.edu/~mjankowski/docs/ele314/labs/index.htm
Bye, Mariusz
>>> On 1/18/2006 at 2:55 am, in message <dqks93$ptf$1 at smc.vnet.net>, bd
satish<bdsatish at gmail.com> wrote:
> Hi buddies,
>
> Which command (in MATHEMATICA) should I use to evaluate the
> so-called Discrete-Time Fourier Series ?
>
> Another bug: ---
>
>
> The command FOURIER[ { list } ] finds the FFT (Fast Fourier
> Transform) of a list. How can I evaluate FFT symbolically (instead of
> a
> numerical list ) ?
>
> Specifically ,
>
> find the FFT of a^n * UnitStep[n] , n belongs to integer
> and Abs[a] < 1