MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 003 Signals and Systems Spring 2004 Tutorial 2 Tuesday February 17 2004 Announcements Problem set 2 is due this Friday There are no tutorials this week but we will provide this handout and TAs will hold extra o ce hours during the week Today s Agenda DT Convolution Calculating the DT convolution CT Convolution Calculating the CT convolution LTI System Properties Impulse response Commutativity distributivity associativity and time shift E ect of stability on commutativity Causality and stability Signal Properties vs System Properties Singularity Functions 25 1 DT Convolution In class we saw how to write any DT signal as the superposition sum of scaled shifted impulses using the sifting property of the DT unit impulse x n x k n k k We de ne the unit impulse response h n of a DT system H as the output when the input is the unit impulse n If the system H is LTI then the output y n for any arbitrary input x n is y n H x n H x k n k k x k H n k by linearity k k k y n x k H n shifted by k x k h n shifted by k x k h n k by time invariance apply system H simplify k So the output y n is the convolution of the input x n and the unit impulse response h n of the LTI system We write this as y n x n h n One useful sanity check when performing DT convolutions is that a signal of length m convolved with a signal of length n produces a signal with no more than m n 1 non zero terms 1 1 Calculating the DT convolution Calculating the convolution of two DT signals can be a rather tedious process Consider the following example Given x n n 1 2 n 3 n 1 and h n n 4 n 1 2 n 2 n 3 calculate y n x n h n We can calculate the convolution by applying the convolution sum directly y n x k h n k k 26 Since x n is nonzero for n 1 0 1 y n 1 x k h n k k 1 y 1 x 1 h 0 1 y 0 x 1 h 1 x 0 h 0 4 2 2 y 1 x 1 h 2 x 0 h 1 x 1 h 0 2 8 3 7 y 2 x 1 h 3 x 0 h 2 x 1 h 1 1 4 12 17 y 3 x 0 h 3 x 1 h 2 2 6 8 y 4 x 1 h 3 3 Thus y n n 1 2 n 7 n 1 17 n 2 8 n 3 3 n 4 It is sometimes quicker and less error prone to use the following trick when calculating the convolution of two finite length DT signals First we form the following table x n h n 1 2 3 1 1 2 3 4 4 8 12 2 2 4 6 1 1 2 3 The table consists of the values of x n in the rst column and the values of h n in the top row The elements in the interior of the table are formed by multiplying the corresponding elements in the rst column and rst row Note that the underlined entries correspond to the element at zero i e x 0 and h 0 The double underlined element corresponds to the entry formed by multiplying the two underlined elements i e x 0 h 0 Now we can simply form y n by summing diagonally Using sequence notation y n 1 4 2 2 8 3 1 4 12 2 6 3 1 2 7 17 8 3 Note again that we have underlined the element resulting from a sum containing the double underlined element which corresponds to y 0 This result for y n is equivalent to the one above 27 Problem 2 1 Evaluate the following discrete time convolution sums given below n a y n cos 12 n 12 u n 2 b y n u n p 0 n 2p c y n x n h n where x n and h n are given below 2 1 r 2 r x n 2 r 1 0 1 r 2 r 1 1 1 h n r 2 3 r 4 r n 5 r 4 1 r r 3 r 2 0 1 r r 2 Workspace 28 1 1 r 2 n 2 CT Convolution Similarly we can express CT signals as the superposition sum of scaled shifted impulses using the sifting property of the CT unit impulse x t x t d We likewise de ne the unit impulse response h t of a CT system H as the output when the input is the unit impulse t If the system H is LTI then the output y n for any arbitrary input x n is y t x t h t d or y t x t h t One useful sanity check when performing CT convolutions is that a signal of length t1 convolved with a signal of length t2 produces a signal of length no more than t1 t2 2 1 Calculating the CT convolution To graphically compute the convolution integral x t h t use the following steps 1 Plot x and h 2 Select which signal to ip say h In general the signal of shorter duration is the one you should ip to make the computation easier 3 Flip h about the vertical axis which gives us h 4 Plot h t on the axis This is just h shifted to the RIGHT by t Note that in order to shift h to the right by t we replace by t which gives us h t h t 5 Identify di erent regions of overlap as t is varied from to where there are breaks in either function These correspond to discontinuities and regions where the mathematical expressions describing the functions change 6 Multiply both functions and integrate them for each of the regions identi ed above Make sure to use the correct limits for the integrals 29 Problem 2 2 Evaluate the following continuous time convolution integrals given below a y t u t 1 u t 2 b y t u t 2 u t 1 u t 2 c y t x t h t where x t and h t are given below x t y t 2 2 1 1 1 1 0 1 2 3 t 2 0 2 Workspace 30 1 2 t 3 LTI System Properties 3 1 Impulse response The impulse response completely characterizes an LTI system i e there is a one to one correspondence between the set of impulse responses and the set of LTI systems In fact we can even define or identify an LTI systems by its impulse response h t If we want to nd h t we …
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