2. Derive the Fourier series expansion for each of the following discrete-time signals:
e) Let n] denote the discrete-time unit-step function. Derive the discrete-time Fourier transform of the following signals ytn) - 5 0.25" cos 2xm 5 marks]
Find the discrete-time Fourier Series for the following periodic signals: 3. 4 cos 2.4n n + 2 sin 3.2n n x[n] a. xn 0 12 15 6 b. xn 2N No 2N C.
2) Find the Fourier series of the following continuous time signals S: 2? C. Isin(nt)L
Consider the discrete-time periodic signal n- 2 (a) Determine the Discrete-Time Fourier Series (DTFS) coefficients ak of X[n]. (b) Suppose that x[n] is the input to a discrete-time LTI system with impulse response hnuln - ]. Determine the Fourier series coefficients of the output yn. Hint: Recall that ejIn s an eigenfunction of an LTI system and that the response of the system to it is H(Q)ejfhn, where H(Q)-? h[n]e-jin
9. Find the Fourier series coefficients and Fourier transform for each of the following signals: a) x(t)= sin(10nt+ b) x(t) = t) 1 + cos(2π cos (2rt S2n
2. For each of the periodic signals shown below, (a) Compute the exponential Fourier-series. (b) Sketch the magnitude and phase spectra for - 55ns5. Compute the relative error due to truncation when only 11 terms (-5S S5) in the series expansion are kept. *** WAN_ + 1 + 3 + 5 0 2 4 *_ _ 0 2 4 6
LTI Systems and Discrete-Time Fourier Series-1 Problem Statement Consider a causal discrete-time LTI system whose input r[n] and output yinl are related by the following equation: Find the Fourier series representation of the output y[n] for (b) ncos()
5. (20points) What is the discrete time Fourier series of 2T(n-2) x = 10 sin with Mo = 12 (time period)
5. (20points) What is the discrete time Fourier series of 2T(n-2) x = 10 sin with Mo = 12 (time period)
3.11-For each of the following signals compute the complex exponential Fourier series by using trigonometric identities,and then sketch the amplitude and phase spectra for all values of k (a) x(t)-cos(5t-π/4) (b) x(t) sint+ cos t 756 Chapter & The Series and fourier Translorm 023 4 5 ibi FIGURE Pa P33 3.13 Problems 157 in 0 14 12 3 I) ain FIGURE ,3.3 (antísndj (c) sti)-cos(1-1) + sin(,-%) 3.12. Determine the exponential Fourier series tor the Following periodic signals
3.11-For each...
Compute the Discrete-Time Fourier Transform analytically for the following signals and plot the absolute values and the phase of the DTFT from-2π to 2π x[n] αηυ[n] for α-0.7 and 0.3 x[n]-δ[n-r] for τ-2 and 3 xInrk], for r -2 and 3 a. b. C. Please show your work step by step and include the formula for finding the absolute value of DTFT and the phase of DTFT.