Find ao, an's, b's, C's and D's. Also, evaluate an, bn, Cn and D' for n=3 (find a3, b3, C3 and D and _3.) n

Problem 2 If s signal, x(t), is transmitted over a channel that produces a received signal y(t) where y(t) = 2 x(t-0.003 μs) +0.004-x(t -0.005 μs) a) For the case of this input x(t) = x₁ (t) to the system above what would be the output of this channel? x₁(t) → y₁ (t), Find an expression for y₁ (t). input is x₁(t) = + 1.1-cos(2n500t)? (write an expression for output) b) For the case of this input x(t) = x₂(t) = 8(t), an impulse, to the same system again, what would be the response, y(t) = y₂(t), x₂ (t)→ y₂(t), Find an expression for y₂(t). c) Is this system LTI? d) Does this channel/system provide distortionless transmission? e) What would be the transfer function, H(f), for this system?

Exercise 1. FOURIER SERIES REPRESENTATION OF PERIODIC SIGNAL (i) Determine the Fourier series coefficients a, for the following signal, and write down the signal z(t) in terms of its Fourier series coefficients: (ii) Repeat (i) for the following signal:

Exercise 2. CONTINUOUS-TIME SIGNAL RECOVERED FROM FOURIER COEFFICIENTS Consider two periodic signals, 7₁ (t) and 1₂(t). ₁ (t) has period 2 and its Fourier series coefficients are al = a₁ = 2 and a = 0 for all k +1. 1₂(t) has period 3 and its Fourier series coefficients are a₁ =j,a-1-j and ax = 0 for all k + +1. (i) Plot x1(t) and x₂(t). (ii) Consider a new signal y(t) = £₁(t) +12(t). Find the Fourier series representation of the signal y(t).

Determine the signal r(t) for the given Fourier series coefficients given that the period T = 4. (You can apply Fourier series properties to examples done in the lectures rather than re-computing the entire problem from scratch.)

Consider the following continuous-time signals: r(t) = cos(4πt) y(t) = sin(Ant) (i) Determine the Fourier series coefficients at of r(t) and the Fourier series coefficients be of y(t). (ii) Consider the function z(t) = x(t)y(t). Using the multiplication property of the continuous-time Fourier Series, determine the Fourier series coefficients c of z(t). (iii) Obtain the Fourier coefficients in (ii) an alternate way: expand z(t) using a trigonometric identity and then compute c from z(t) directly. Show that your answer matches that found in part (ii).

A CT period signal x(t) is real valued and has a fundamental period of T = 6. The nonzero

Exercise -5: Design a 2nd order bandpass digital filter with center frequency at f=1.6kHz and a 3-dB bandwidth of 400Hz.

Determine the Fourier series coefficients of the signal x(t) that is periodic with period

Let be a periodic signal with fundamenatl period T = 2 and Fourier coefficients ak/ Determine the value