1 x(t) is a continuous-time real sinal. Let (t) and (t) the even and the odd...
Q1) For the continuous time signal below, x(t)=1+t (a) Determine the even and odd parts of the signal. (b) Sketch the signal from t = -3 to t = 3. (c) Explain why the signal does not possess BIBO stability.
A) Assume that a continuous-time aperiodic signal x(t) is real. Prove that the spectrum X(jω) satisfies the following property: X(jω) =X(−jω)∗ where ∗ denotes conjugation. B) Assume that a continuous-time aperiodic signal x(t) is real and even. Prove that the spectrum X(jω) is real and even.
1. Determine if v(t) = t^2 (t-squared) is an even or odd function. Show your proof. 2. Determine if v(t) = t^2 (t-squared) is an even or odd function. Show your proof. 3.A sinusoidal signal has a period of 10 ms. What is the angular frequency? 4.A sinusoidal signal has a period of 10 ms. What is the angular frequency?
151 4.8 Problems 4.17 The autocorrelation function Txx(t) of a real analog signal x (t) is defined by xx(t) x(t)x (r-t) dt, (4.73) obtained from Eq. (4.70) by replacing y(t) with x (t). Thus the function is a cross-correlation of x(t) with itself. One application of the autocorrelation function is to detect the period of a periodic signal that has been corrupted by noise. Show that rxx(t) is an even function of r.
151 4.8 Problems 4.17 The autocorrelation function...
Problem 3: a) Show that is f(t) is an even, real valued periodic function of time with period To, then 0 f(t)dt ao = T. Jo b) Show that is f(t) is an odd, real valued periodic function of time with period To, then an-0 f (t) sin(nwot)dt
(a) Let x(t) be a continuous-time signal known to have a first derivative ct) that is a smooth, continuous function over all t in (-00,00). Then the integral [ [x(t) – e(t – 7)][8(t – 3) + 6(t – 10)] dt evaluates to which of the following expressions: 1. x(t)8(t – 3) 2. x(3) 3. x(3) – č(-4) + x(10) – č(3) 4. x(3) – 3(-4) (b) A continuous-time dynamic system is described by the differential equation dyſt) + 4y(t)...
1. Show that if x(t) is an even function of t, then X(jw)2 (t) cos(wt) dt and if r(t) is an odd function of t, then X(jw)2j (t) sin(wt) dt
Let P1 = 1 If x is odd then Px+1 = 2Px If x is even then Px+1 = 2Px +1 Show this is true and solve it: 2Px+1 + 2Px+1 +1 = Px+2
(a) Determine algebraically whether the functions below are even, odd or neither. i. r+6 f(x)=- r-r? (2 marks) ii. f(x) = 2x sinx (2 marks) (b) A periodic function is defined by: f(x) = 4-x?, -25x52, f(x+4)= f(x) i. Sketch the graph of the function over -10<x<10. (4 marks) ii. Based on result in (i), identify whether the function is even or odd. Give your reason. (2 marks) ii. Calculate the Fourier series expansion of f(x). (12 marks) (c) An...
Proof
Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous functions on a compact subset S of R such that (1) for each x € S, the sequenсe {fn(x)}o is bounded and топotone, and (ii) the function x lim,0 fn(x) is continuous on S Then f Remark that the result is not always true without the monotonicity of item (i) Šn=0 lim fn uniformly on S
Theorem 65.6 (a generalization of Dini's theorem) Let...