2. Determine whether the following signals are periodic. If...

2. Determine whether the following signals are periodic. If they are periodic, find the fundamental period, frequency, and angular frequency. (a) (x(t) = cos(3pi t) + 1) (b) (x[n] = e^{jpi n} sin(frac{pi n}{2})) (c) (x[n] = sum_{k=-infty}^{infty} delta[n - 3k] + delta[n - 2k]) (d) (x(t) = sum_{n=0}^{infty} 2^{-n} delta(t - 2n)) (e) (x[n] = cos(n) + sin(n))

Transcript

00:01 Here we have to determine whether the signals are periodic or not.

00:03 So first we are considering here that is x of t is equals to cos of 3 pi of t plus 1.

00:09 So this is the first period here.

00:11 So here we can say that this signal from here is periodic because dc does not affect periodically here.

00:17 So we can say that f from here is equals to 3 of pi that is divided by 2 .25 that is equals to 1 .5 so we can say that t0 from here is equals to 2 divided by 5 second from here where omega become equals to 3 pi that is in radian per second.

00:35 So we can say that this signal is periodic from here.

00:38 Hence the answer to the part a of the question.

00:40 Now we are considering about the part b where x of n from here is equals to e raised to the power j pi of n multiplied by the sin of pi of n that is so this is given from here.

00:50 So we can write it as e raised to the power j of pi of n that is divided by 2 of j multiplied by e raised to the power j pi that is divided by 2 of n minus e raised to the power minus j pi divided by 2 which is multiplied by n is here.

01:04 So this from here is equals to 1 divided by 2j multiplied by e raised to the power j that is 3 of pi which is divided by 2 multiplied by n minus e raised to the power j pi divided by 2 that is further multiplied by n.

01:15 So again we can say that this is periodic from here.

01:19 So the value of t1 is equals to 3 of pi that is divided by 2 multiplied by 2 of pi from here that is equals to 3 which is divided by the 4 and t2 become equals to pi that is divided by 2 multiplied by 2 of pi that is equals to 1 divided by 4...

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