The signals xa[n] and xb[n] are periodic. Find the Fourier coefficients of the signals.

Name: The signals xa[n] and xb[n] are periodic. Find the Fourier coefficients of the signals.
Uploaded: 2022-12-12T12:33:42-08:00
Duration: 6 min 22 s
Channel: Sri K
Description: The signals xa[n] and xb[n] are periodic. Find the Fourier coefficients of the signals.

Step 1: Calculate the period of the signals xa[n] and xb[n] by finding the smallest positive int

The signals xa[n] and xb[n] are periodic. Find the Fourier...

Transcript

00:01 In this question we need to find out the periodic signals.

00:04 Okay, so we have to find out what is the periodic signals denoted by x of n with fundamental period.

00:15 We have been given an equals to 8 is a fundamental period and we have been also given the fourier coefficient which is k equals to ck equal to cost of k pi divided by four plus sign of three k pi divided by four this is what we have been given we have to find out the periodic signal so let's start with this so since we know that the periodic signal x n is given by summation k going from zero to seven because the fundamental period is given to be 8.

00:56 So k is going from 0 to 7, k, e raise 2, j times of 2 pi divided by 8, multiplied with nk, correct? and also the fourier coefficient, if i try to find k, which is given to be e raised to j, and this is nothing but 2 pi divided by 8 of p times of k.

01:26 So hence if i try to write down what is this value of xn periodic signal, it will be summation k going from 0 to 7, e raise 2 j times 2 pi by 8, and in nk and p k is common.

01:45 So it will become just n plus p, correct? which is equal to 8 when p is equal to n sorry p equals to minus n and it is 0 when p is not equals to minus n this is our observation correct so since we have c k equals to cos of k pi divided by 4 plus i times sine of 3 k pi divided by 4 so what i can do here k i will rewrite in the fourier coefficient i will just simplify this fourier coefficient okay let's see what we are going to get e raise to j here i will write 2 pi divided by 8 times of k plus e raise 2 minus j 2 pi divided by 8 times of k correct then plus 1 divided by 2 j 2 j 2 .5 divided by 8 times of k j times e raise 2 j and here i will get 6 pi divided by 8 because we have sign of 3 k by 4 okay so due to that i am writing 6k divided by 8 k minus e raise 2 minus j 3 sorry 6 pi divided by 8 times of k this is what i am going to get correct now we have to just simplify this and let us see what we are going to get.

03:24 So this will give us if we try to simplify it one step ahead.

03:30 So i am going to get this to be equal to summation k equals to 0 to 7 1 divided by 2...

Question

The signals $x_a[n]$ and $x_b[n]$ are periodic. Find the Fourier coefficients of the signals.

Question

The signals $x_a[n]$ and $x_b[n]$ are periodic. Find the Fourier coefficients of the signals.

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