6. (a) The signal y(t) is defined as follows: ' y(t) = r(r)dT Suppose that (t)...
(b) (5 pts) Unit Impulse. Suppose we have an impulse train signal h(t)-Σ δ(t-nT). Given an arbitrary signal r(t), find r(t)h(t) and (t) h(t) in terms of r(t) Show that r(t)h(t)-Σ r(nT)δ(t-nT) and r(t) * h(t)-Σ r(t-nT) (b) (5 pts) Find the Fourier Transform of r(t) (t 2n). Hint: Find wo and the Fourier series coefjicients then use the Fourier Transform property for periodic signals. (b) (5 pts) Unit Impulse. Suppose we have an impulse train signal h(t)-Σ δ(t-nT). Given...
(a) Let the correlation be defined as r (t) x(T) y (tT) dT T Express R jw= F{r (t)} in terms of X (jw) and Y (jw), the Fourier transform of x (t) and y (t) respectively. (b) Suppose (t) = y (t) = e-H. Find R (jw) using frequency domain properties and the relationship derived in (a) extra Find R (jw) by evaluating the convolution integral in the time domain to get r (t) and then doing the FT.
Suppose that a period signal x(t) is defined by the plot below No idea how they're getting the answers on this page. PROBLEM Spring-2020-Q.2.2: Suppose that a periodic signal r(t) is defined by the plot below (only the section - 12<t < 12 is shown, although the signal is define for -0 <t<oo). 1ERA -12 -8 4 8 12 (a) [2 points] Determine the fundamental frequency of r(t) in radians/second. wo = 7/6 rad/s (b) [4 points] Since r(t) is...
Consider a signal x(t) = e-tu(t), and the signal y(t) below: dx(t) y(t) = 3e-33+ z(t – 5) + 5* dt Va) What is X(jw), the Fourier transform of æ(t)? b) Find the phase of the complex number X(j1). c) Find Y(jw), the Fourier transform of y(t). d) Find the magnitude of the complex number Y(j1).
In each step to follow, the signals h(t), a(t), and y(t) denote respectively the impulse response, input, and output of a continuous-time LTI system. Accordingly, H(w), X(w) and Y(w) denote their Fourier transforms. Hint: Carefully consider for each step whether to work in the time domain or frequency domain. (b) (25 points) On the axes below, provide a clearly labeled sketch of y(t) for all t given Σ H(w)-( ) sine? (w/8) j2Tt r(t)-e δ(t-n/2) and with sinc(t) = sin(t)/t...
Consider the DT LTI system defined by the mpulse response h[n] = ?[n] The input to this system is the signal rn: ?[n-1) (a) Sketch h[n] and r[n] (b) Determine the output of the systern, ylnj, using convolution. Sketch y[n] (c) Determine the DTFTs H(e) and X(e. Make fully-labeled sketches of the magni- tudes of these DTFTs (d) Recall that the discrete Fourier transform (DFT) is simply defined as samples of the discrete-time Fourier transform (DTFT). Compute the 4-point (N-4)...
HW 11.5 Consider the periodic "square wave" signal defined by x(t)- u(t - 4k) - u(t - 2-4k) (a) Sketch x(t) (b) Sketch g(t) = x(t)-0.5 (c) Sketch |x(jw)|. Hint: First determine the Fourier series expansion of x() (d) Sketch IG(Go) HW 11.5 Consider the periodic "square wave" signal defined by x(t)- u(t - 4k) - u(t - 2-4k) (a) Sketch x(t) (b) Sketch g(t) = x(t)-0.5 (c) Sketch |x(jw)|. Hint: First determine the Fourier series expansion of x() (d)...
Suppose the signal x[n] = δ[n-1] + cos(nn/6+ π/4) is input to the LTI system described by the equation y[n][n] +r|n - 0.5yn -. Determine a closed-form expression for yn] Suppose the signal x[n] = δ[n-1] + cos(nn/6+ π/4) is input to the LTI system described by the equation y[n][n] +r|n - 0.5yn -. Determine a closed-form expression for yn]
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
6- A contiuous-time periodic signal x(t) is given graphically below. (a) Determine the exponential Fourier coefficients for k+oo a ()-ΣGeko, k-oo where c is given by T/2 1 (t)ek dt J-T/2 Ck= T (b) r(t) is applied as an input to an LTI system whose frequency response is H(ju)=2 sin(w Determine the corresponding output y(t) (e) Sketch y(t). Be sure to mark the axes properly -JT 6- A contiuous-time periodic signal x(t) is given graphically below. (a) Determine the exponential...