LECTURE 18: THE JOY OF CONVOLUTION

Transcription

1 LECTURE 18: THE JOY OF CONVOLUTION Part 1: INTRO 1. Just because no one understands you, doesn t mean you re an artist. Rich Hebda. I am reminded of this quote every time I talk about convolution. 2. WORDS: CONVOLUTED: folded in curved or tortuous windings. CONVOLVE: to role together to writh (in agony over HW problems). PRESCRIPTION: read the notes first. 3. RECALL LECTURE 1. Now you will understand why we avoided staying in the t-world and did intensive counseling in the s-world to solve our circuits problems. Input Signal (voltage/current source) Circuit /Model h(t) Impulse Response Output Signal (voltage/current)!!! L Input Signal L Circuit /Model H(s) Transfer Function L Output Signal

2 Lecture 17 Sp 18 2 R. A. DeCarlo COMMENT: Convolution allows us to traverse the top part of the diagram for a much larger class of signals than the one-sided Laplace transform what would one expect since the transform is SOOOO one-sided. DEFINITION: Let f (t) be an input signal possibly nonzero for t < and y(t) the circuit output signal. Suppose h(t) is the impulse response of the circuit. THEN y(t) is the convolution of h(t) with f (t), denoted as y(t) = h(t)* f (t) = f (t)* h(t), and defined by the integral y(t) = h(t τ ) f (τ )dτ = h(τ ) f (t τ )dτ 4. What is the meaning of the convolution integral? y(t) = h(t τ ) f (τ )dτ (i) t is the time at which the output y(t) is observed/measured. (ii) τ is the time when a piece of the input signal is applied. (iii) t τ is the time that has elapsed between the application of the input signal and the observation/measurement of the output signal.

3 Lecture 17 Sp 18 3 R. A. DeCarlo (iv) h(t τ ) measures the effect that a nano piece of the input signal at time τ HAS on the output at time t, i.e., the effect at time t due to the input f (τ ) is precisely h(t τ ) f (τ ). (v) By superposition, the integral sums up each and every little crumb of a contribution. Comment: Imagine ants climbing out of their nest at the dining hall and crawling along the broken road of glass, cake crumbs, spills, footprints, and dirt. Some ants get lost along the way. But every student is but a northern star, a sign, pointing straight to the loving arms of the outside sunshine and warm temperatures (somewhere in Florida). So only a handful, some fraction, say h(t τ ), of the original nest escapees wipe their brow (and antennae) and push on through as they march on home into. (Apologies to Rascal Flats.) PART 2: GENERAL EXAMPLES Example 1. y(t) = h(t)*[k δ (t T )]. From the sifting property of the delta function, y(t) = h(t)*[k δ (t T )] = K h(t T ).

4 Lecture 17 Sp 18 4 R. A. DeCarlo Example 2. Show that r(t) = u(t)*u(t). (VERY IMPPORTANT) Step 1. Write down the convolution integral and make a simple observation. u(t)*u(t) = since u(τ ) = 1 and is nonzero only for τ. u(t τ )u(τ )dτ = u(t τ )dτ Step 2. My dear Watson, consider the case of the negative t, i.e., t <. From step 1, τ. And so, t τ < implies u(t τ ) =, and so for t <, y(t) =. Step 3. My dear Watson, now consider the case of the non-negative t, i.e., t. From step 1, τ. And so, t τ only when τ t and so u(t τ ) = 1 only when τ t in which case: for t, u(t)*u(t) = t u(t τ )dτ = tu(t) = r(t) EOE Student: Hey Professor Ray, a question. Ray: Yes.

5 Lecture 17 Sp 18 5 R. A. DeCarlo Student: Ya know, it is a heck of a lot easier to use the bottom half of your diagram since L{ u(t)*u(t) } = L{ u(t) } L{ u(t) } = 1 s 2 and 1 L 1 Ray: Well, I was just trying to see if you were paying attention. s 2 = r(t). Student: Yeah Right Professor Ray. Now if I were you and you were me, I would have said that I was trying to teach you how to solve a convolution integral even though it is easier to solve the problem in the s-world. You were always telling us how easy it is to solve things in the s-world, right? Ray: Ha ha. Some Properties of Convolution: 1. Commutativety: h* f = f * h 2. Associativity: h*( f * g) = (h* f )* g 3. Distributive: h*( f + g) = h* f + h* g 4. TIME SHIFT PROPERTY (VERY VERY IMPPORTANT): if y(t) = h(t)* f (t) then h( t T 1 )* f ( t T 2 ) = y( t T 1 T 2 ).

6 Lecture 17 Sp 18 6 R. A. DeCarlo Example 3. Compute y(t) = h(t)* f (t) where h(t) and f (t) are shown in the figures below f(t) blue and h(t) red time Strategy: combine the results of Example 2 with the properties of convolution, especially the time shift property. y(t) = h(t)* f (t) = f (t)* h(t) = u(t +1)* 2u(t) u(t 1) u(t 2) u(t 1)* 2u(t) u(t 1) u(t 2) = 2r(t +1) r(t) r(t 1) 2r(t 1) r(t 2) r(t 3) = 2r(t +1) r(t) 3r(t 1) + r(t 2) + r(t 3)

7 Lecture 17 Sp 18 7 R. A. DeCarlo y(t)=h(t)*f(t) Time Example 4. Compute y(t) = h(t)* f (t) where h(t) = u(t) u(t T ), T > and f (t) = u( t), an input from the past. Solution. Step 1. Again y(t) = h(t)* f (t) = f (t)* h(t): Step 2. Small details. y(t) = u(τ t) u(τ ) u(τ T ) dτ u(τ ) u(τ T ) = 1 τ < T otherwise Thus

8 Lecture 17 Sp 18 8 R. A. DeCarlo T y(t) = u(τ t) dτ Step 3. Case 1: t <, τ < T, implies u(τ t) = 1 implies that for t <, y(t) = T dτ = T Step 4. Case 2: t < T, τ < T implies u(τ t) = 1 τ t otherwise y(t) = T t dτ = T t Step 5. Case 3: t > T, τ < T, implies τ < t implies τ t < implies u(τ t) = implies y(t) =. Step 6. Summary: y(t) = T t < T t t < T T t The plot below is obtained by setting T = 1.

9 Lecture 17 Sp 18 9 R. A. DeCarlo u( t)*[u(t) u(t T), T = Time Example 5. Compute y(t) = h(t)* f (t) where h(t) = e t u( t), an impulse response from the first age of middle earth, and f (t) = e t u(t), Step 1. Write down the specific convolution as per the definition of convolution: y(t) = e t τ u(τ t)e τ u(τ ) dτ = e t e 2τ u(τ t) dτ since u(τ ) = for τ <.

10 Lecture 17 Sp 18 1 R. A. DeCarlo Step 2. Break the integral up into cases. Case 1: t <, τ. (i) u(τ t) = 1 and (ii) y(t) = e t e 2τ dτ = e t e 2τ 2 =.5e t, t < Case 2: t, τ : (i) u(τ t) = 1 τ t t > τ and (ii) y(t) = e t e 2τ u(τ t) dτ = e t e 2τ dτ = e t e 2τ 2 t t =.5e t, t Step 3. Summary: y(t) =.5e t t <.5e t t =.5e t u( t) +.5e t u(t)

11 Lecture 17 Sp R. A. DeCarlo The Convolution time Student: Hey Professor Ray, this convolution stuff is like a 5% off sale at Sofa King Cool.