Determination of a signal from its Fourier Coefficients

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Discussion Overview

The discussion revolves around the determination of a discrete-time signal from its Fourier coefficients, specifically focusing on the method of deriving the signal x[n] from the given coefficients ak. Participants explore different approaches to simplify the computation process, especially when dealing with larger periods.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant presents a method for deriving x[n] using a summation of exponential terms based on the Fourier coefficients ak, noting the tedious nature of evaluating the sum for a fundamental period of N=12.
  • Another participant suggests that the original poster should avoid converting to Euler's formula, as it complicates the expression, and instead sum the coefficients directly.
  • A different participant points out that patterns in the coefficients can simplify calculations, using k=4 as an example to illustrate potential simplifications.
  • One participant proposes considering programming a solution to automate the computation, indicating a lack of known mathematical shortcuts for manual calculations.

Areas of Agreement / Disagreement

Participants express differing opinions on the best method to derive x[n], with no consensus on a single effective approach. Some advocate for direct summation while others suggest avoiding Euler's formula.

Contextual Notes

The discussion highlights the potential complexity of calculations for larger periods, with participants acknowledging the tedious nature of the summation process without resolving the best method to simplify it.

Who May Find This Useful

This discussion may be useful for students or practitioners interested in signal processing, particularly those dealing with Fourier analysis and seeking efficient computation methods for discrete-time signals.

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letsa say i have an ak = cos ( k*Pi/4) + sin(3*k*Pi/4), the signal is discrete time, fundamental period N=12.

the way i would derive its x[n] is.. Sum(k=0, to 11 of: 0.5*exp(j*k*Pi/4) *exp(j*k*w*n) + 0.5*exp(-j*k*Pi/4) *exp(j*k*w*n) + (1/2*j)*exp(j*k*3*Pi/4) *exp(j*k*w*n) - (1/2*j)*exp(-j*k*3*Pi/4) *exp(j*k*w*n)

In other words, i expanded ak using eulers, and put it into the formula for finding x[n]... however as u see it turns out to be tedious process to evaluate this sum 12 times..

i was wondering, what is the more effective and QUICK method getting of x[n] by hand?

thanks
 
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You mean X(t) right ?

Anyway, do not convert to euler as you see it looks terrible.

You can just sum it the way you see it.

Another choice would be to invest a little time to simplifying the equation.

For example whe k =4

cos ( k*Pi/4) = -1
sin (3*k*pi/4) =0
Usually this kind of patterns repeat.
 
no actually x[n] because its discrete time... wht if my period would be 40 or more? it would be more tedious then...
 
Well, perhaps you should look into creating a program to do the job. I personally do not know any mathematical acrobatics that can help you.
 
thanks anyway!
 

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