Determination of a signal from its Fourier Coefficients

[ColdStart]

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

 

[╔(σ_σ)╝]

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.

 

[ColdStart]

no actually x[n] because its discrete time... wht if my period would be 40 or more? it would be more tedious then...

 

[╔(σ_σ)╝]

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.

 

[ColdStart]

thanks anyway!