3. Find the Fourier transforms of the signals: (a) f(t) =...

3. Find the Fourier transforms of the signals: (a) $f(t) = te^{-t}$ (b) $g(t) = t^2e^{-t^2}$ (c) $h(t) = tu(t)$ (d) $k(t) = cos(at)$ (e) $l(t) = sin(at)$ (f) $m(t) = e^{-t}sin(at)$ (g) $m(t) = tsin(at)$ (h) $o(t) = \frac{sin(at)}{t}$ (i) $p(t) = e^{-|t|}$

Transcript

00:01 Hello students, in this question we have to calculate the fourier transform of the given function.

00:07 So, in the first part of the question given as the function which is a form of the time t.

00:14 So, this is equals to sin 2t.

00:23 So, the fourier transform function is that is a function of f frequency 1 upon under root 2 pi is the normalization factor exponential minus infinity to infinity x of t into exponential i f t into dt.

00:49 Now, we will substitute the values 1 upon under root 2 pi integration minus infinity to infinity x of t is the sin 2t is given exponential i f t dt.

01:06 So, 1 upon under root 2 pi integration minus infinity to infinity sin 2t we can write exponential i 2t minus exponential minus i 2t upon 2i exponential i f t dt.

01:39 So, this is equals to 1 upon under root 2 pi integration minus infinity to infinity we can take 2i as a common.

01:48 So, 2i here and exponential will be exponential i is common in bracket 2 2 plus f t and minus exponential i minus 2 plus f t dt.

02:19 So, after integration we will get this function is equals to the i under root pi by 2 in bracket delta of f minus 2 minus delta of f plus in the next part of the question x of t is given as cos 2t.

02:49 So, this is will be equal to the again 1 upon under root 2 pi integration minus infinity to infinity cos 2t exponential i f t into dt.

03:05 So, after integration similarly as above is equals to under root pi by 2 in bracket delta of f minus 2 plus delta of f plus in the next part the function is exponential mod of t and sin of 2t.

03:27 So, the fourier transform of above function is not possible because it is not elementary and involving convolution of exponential minus mod t and sin of 2t...

Question

3. Find the Fourier transforms of the signals: (a) $f(t) = te^{-t}$ (b) $g(t) = t^2e^{-t^2}$ (c) $h(t) = tu(t)$ (d) $k(t) = cos(at)$ (e) $l(t) = sin(at)$ (f) $m(t) = e^{-t}sin(at)$ (g) $m(t) = tsin(at)$ (h) $o(t) = \frac{sin(at)}{t}$ (i) $p(t) = e^{-|t|}$

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