3) Find the inverse Fourier transform of X(w) = jw +3 -w2 + j3w + 2...
Find the inverse Fourier transform for the following signals. X(e^jw) = 2 cos(w)
Given that the Fourier transform of x(t) is 3e-jw x(jw) = (1 +ju) find the Fourier transform of the following signals in terms of X (jw). a. y(t) = e'*x(t – 2) b. y(t) = x(-3) c. y(t) = x(t)dt
Find the inverse Fourier Transform of H(jω) shown below: 6(3+j2w) H(W) = (1 + iw)(4 + jw)(2 + jw) Answer: h(t) = (2e-+ 3e-2t – 5e-4t)u(t)
Useful Formula: Fourier Transform: F[f(t)] = F(w) sof(t)e-jw dt Inverse Fourier Transform: F-1[F(w)] = f (t) = 24., F(w)ejwidw Time Transformation property of Fourier Transform: f(at – to). FC)e=itoch Laplace Transform: L[f(t)] = F(s) = $© f(t)e-st dt Shifting property: L[f(t – to)u(t – to)] = e-toSF(s) e [tuce) = 1 and c [u(e) = ) Using the convolution property of Fourier Transform to find the following convolution: sinc(t) * sinc (4t) [Hint: sinc(t) or rect(w/2)] TC .
2. Calculate the inverse Fourier transform of X(cfw) = {2 2j 0 <W <T -2j -n<w < 3. Given that x[n] has Fourier transform X(@j®), express the Fourier transforms of the following signals in terms of X(el“) using the discrete-time Fourier transform properties. (a) x1[n] = x[1 – n] + x[-1 - n] (b) x2 [n] = x*[-n] + x[n]
x(t) has the fourier transform x(jw) show dx(t)/dt has the fourier transform jw x(jw)
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts) Find2X(jw)dw. Hint: Apply the definition of the inverse Fourier transform formula, and you can also recall the time shift property for Fourier Transform. (c) (5 pts) Fourier Series. Consider the periodic signal r(t) below: 1 x(t) 1 -2 ·1/4 Transform r(t) into its Fourier Series...
fourier analysis 2. (a) Find the Fourier sine transform of b) Write f(x) as an inverse sine transform Hint: Don't directly calculate F,[f (x)(w). Begin with showing the representation sin wxdw. x >0 ㄧㄨ and then interchanging x and w in the representation. Now look at it carefully, what does the equation tell you?
(b) (2 pts) (t) is given as r(t) e sin(t) Find X(jw). Show that X(jw) = 25 + (w- 1)225(w+1)2 (c) (4 pts) x(t) is given as x(t)-π inc(t) cos(nt). Find X(jw) (d) (4 pts) 2(t) is given as 2(t) e Áil+ 3) + e' ỗ(t-3). Find X (jw). Simplify the answer as (e) (4 pts) 2(t) is given as r(t) = rect(2(t )) reetgehj)). Hint: use Fourier Transform pair: sine(t)艹rect( ) much as possible Find X(jw). Simplify the answer...
A sequence has the discrete-time Fourier transform 1 - a2 X(e) ae-jw)2(1- aejw) la| < 1 (a) Find the sequence r[n] (b) Calculate X(eju)cos(w)dw/27