4- Discrete-Time Fourier Transform of a sequence...

4- Discrete-Time Fourier Transform of a sequence is: X(e^{j?}) = frac{1 - a^2}{(1 - ae^{-j?})(1 - ae^{j?})} a) Find sequence x[n] b) Compute the following integral: frac{1}{2pi} int_{-pi}^{pi} X(e^{jomega}) cos(omega)domega

Transcript

00:01 In this question we are given a discrete time fourier transform of a sequence and which is given by this xe to the power j omega and now we need to first find the sequence x and then we need to compute the integral the given integral so first of all let us compute the sequence for this discrete time fourier transform so this transform can be written as x of e j omega we are are doing the part a so this can be written as 1 minus a square and multiplying both of them we will get 1 minus a e to the power j omega minus ae to the power minus j omega plus a square and then we can write it as 1 minus a square over 1 minus if we uh if we take this as common and multiply and divide it by multiply and divide it by 2 so we will get 1 minus 2a e to the power j omega plus e to the power minus j omega over 2 plus a square and now we know that e to the power j omega plus e to the power minus j omega over 2 is nothing but equal to cos omega so this will become 1 minus a square over 1 minus 2a cos omega plus a square and this is 1 minus a square and this can be written as x of e to the power j omega can be written as 1 over 1 minus a e to the power minus j omega plus a to the power j omega over 1 minus a e to the power j omega using this problem this so now from properties of fourier transform from properties of fourier transform we have a to the power n mu n this is transformed to 1 over 1 minus a e to the power minus j omega and also a to the power minus n mu of minus and minus 1 will transform to ae to the power j omega over 1 minus a 80 the power j omega so applying these two transformation in in obtained one we will get x n is equal to a to the power n mu n plus a to the power minus n mu of minus n minus one or this can be also written as x n is equal to a to the power n when n is greater than 0 and a to a to the power n when n is greater than 0 and a to the power minus n when n is less or equal to minus 1 or we can say that x n is equal to a to the…

Question

4- Discrete-Time Fourier Transform of a sequence is: X(e^{j?}) = frac{1 - a^2}{(1 - ae^{-j?})(1 - ae^{j?})} a) Find sequence x[n] b) Compute the following integral: frac{1}{2pi} int_{-pi}^{pi} X(e^{jomega}) cos(omega)domega

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