ENGN 1570 Fall 2014 Homework 3 Example 1.2 Problem 1 We are given the signal x(t): x(t) 1 −1 0 1 2 3 4 t 5 Let us transform the given x(t) to x(−2t + 6). We need to use all three types of transformations (a shift, a scale and a flip), but what in what order shall we do them? How do we do it? The following guide explains: For the signal x(t) above. Sketch x ∗ h when (a) What to do when you need to do multiple transformations: 0 t<0 h(t) =of 2 0 < t < 1 • We can think of cascaded transformations as repeated substitutions the independent variable t. 0 t>1 • We can do the transformations in any order. However, the amount and direction of the shift depends on whether it is performed before or after the reversal and the scale. To demonstrate these principles, let’s do the transformation in three different orders. • x(t) (b) h(t) = δ(t) + δ(t − Compress 1) by 2 Reverse Advance by 6 −→ x(t + 6) −→ x(−t + 6) −→ x(−2t + 6). Reverse −→ x(2(−t + 3)). Problem 2 • x(t) Compress by 2 −→ x(2t) Advance by 3 −→ x(2(t + 3)) Let Ch be the system that has output x ∗ h on input x. For each of the choices for h below, determine if Ch is invertible. If so give the signal h0 such that Ch0 is the inverse of Ch . If not • x(t) Delay bysignals 6 Compress by 2 give two that map output signal. x(−t) input −→ x(−(t − 6)) −→ to the x(−(2tsame − 6)). Reverse −→ (a) h(t) = e−j2t (b) h(t) = jδ(t) (c) h(t) = u(t) + u(t − 5) (u is the unit step function). 9 1 Problem 3 Let y = x ∗ h. Let y 0 , x0 and h0 denote y, x and h delayed by 1. Determine which of the statements below are true and which are false. Justify your answers. (a) If x[n] = 0 for n < N1 and h[n] = 0 for n < N2 then y[n] = 0 for n < N1 + N2 . (b) If x[n] = 0 for n > N1 and h[n] = 0 for n > N2 then y[n] = 0 for n > N1 + N2 . (c) y 0 = x0 ∗ h0 (d) y 0 = x0 ∗ h (e) y 0 = x ∗ h0 Problem 4 NOTE: Part (a) of this problem is the same as Problem 6 from Homework 2 but with more non-zero values in the definition of h. One important use of inverse systems is in situations in which one wishes to remove distortions of some type. A good example is the problem of removing echoes from acoustic signals. For example if a room has echo then an initial acoustic impuse will be followed by attenuated versions of the sound at regularly spaced intervals. A model for this is an LTI system with impuse response given by a train of impuses ∞ X hk δ(t − kT ) h(t) = k=0 Here the echos occur T seconds apart, and hk represents the gain factor on the k-th echo. We can think of T and the hk ’s as parameters describing the acoustics of a room. Suppose that x(t) represents an original acoustic signal and that y(t) = x(t) ∗ h(t) is the signal recorded by a microphone. Note that y(t) is the superposition of delayed versions of x(t), scaled by the coefficients hk . The system defined by convolution with h is invertible, therefore we can remove the echos from y(t) by convolving it with another signal g(t). The signal g(t) is also a train of impulses g(t) = ∞ X k=0 gk δ(t − kT ) 2 (a) Suppose that h0 = 1, h1 = 1/2, h2 = 1/4, and hi = 0 for all i ≥ 3. Determine the values for the gk ’s in this case. (b) With the hk ’s defined in part (a), is Ch stable? What about Cg ? Justify your answer. Problem 5 It is often important to transform between continuous and discrete signals. We can define a discrete signal from a continuous signal by “sampling” the continuous signal at integer times. The problem of defining a continuous signal y(t) from a discrete signal x[n] is more complicated because we have to fill in the missing values. A common approach involve using spline basis functions. One of the simplest spline functions is the “triangular shaped” signal defined by 0 t < −1 t + 1 −1 < t < 0 h(t) = 1−t 0<t<1 0 t>1 (a) Sketch h(t) Let x[n] be a discrete signal. We can define a continuous signal y(t) by taking linear combinations of shifted and scaled versions of h(t). y(t) = ∞ X k=−∞ x[k]h(t − k) Sketch the resulting y(t) for the following cases: (b) x[n] = 2 (c) x[n] = n (d) 3 n=0 x[n] = 0 n= 6 0 3 (e) 0 5 2 x[n] = 2 −1 0 n≤0 n=1 n=2 n=3 n=4 n≥5 (f ) In general, what can you say about the resulting y(t) at integer values for t? What about non-integer values for t? 4
© Copyright 2024 ExpyDoc
