Chap3StudentSolutions

21 sketch these functions a g t rect 4 t b g

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: nction was defined as it was, so it could simply be the convolution of a unit rectangle with itself. This convolution can also be done analytically. For t < −1 , in the range of integration, − 1 1 < τ < , the rect function is zero and the 2 2 convolution integral is zero. For t > 1, in the range of integration, − 1 1 < τ < , the rect function is zero and the 2 2 convolution integral is zero. For −1 < t < 0 . Since the rect function is even we can say that rect ( t − τ ) = rect (τ − t ) . 1 1 This is a rectangle extending in τ from t − to t + . For t’s in the range, −1 < t < 0 , 2 2 1 1 1 t − is always less than or equal to the lower limit, τ = − , so the integral is from − to 2 2 2 1 t+ . 2 t+ g (t ) = 1 2 ∫ rect (τ − t ) dτ − 1 2 This is simply the accumulation of the area under a rectangle and therefore increases linearly from a minimum of zero for t = −1 to a maximum of one for t = 0 . Solutions 3-18 M. J. Roberts - 8/16/04 1 1 to t + . For t’s in the 2 2 1 1 range, 0 < t < 1 , t + is always greater than or equal to the upper limit, τ = , so the 2 2 1 1 integral is from t − to . 2 2 For 0 < t < 1 . This is also rectangle extending in τ from t − g (t ) = 1 2 ∫ rect (τ − t ) dτ t− 1 2 This is also the accumulation of the area under a rectangle and decreases linearly from a maximum of one for t = 0 to a minimum of zero for t = 1. t (b) g( t) = rect ( t) ∗ rect 2 This convolution is easily done graphically. t (c) g(t ) = rect(t − 1) ∗ rect 2 (d) g( t) = [rect ( t − 5) + rect ( t + 5)] ∗ [rect ( t − 4 ) + rect ( t + 4 )] Break this convolution down into the sum of four simpler convolutions. 21. Sketch these functions. (a) g( t) = rect ( 4 t) (b) g( t) = rect(4 t) ∗ 4δ ( t) (c) g( t) = rect ( 4 t) ∗ 4δ ( t − 2) (d) g( t) = rect ( 4 t) ∗ 4δ (2 t) Don’t forget the scaling property of the CT impulse. (e) g( t) = rect ( 4 t) ∗ comb( t) Convolution with a comb is relatively easy because it is simply convolution with a periodic sequence of impulses. g(t) 1 ... -2 -1 -1 1 88 ... 1 t Solutions 3-19 M. J. Roberts - 8/16/04 (f) g( t) = rect ( 4 t) ∗ comb( t − 1) This result is identical to the result of part (e). (g) g( t) = rect ( 4 t) ∗ comb(2 t) Don’t forget the scaling property of the CT impulses in the comb function. The average value of g ( t ) is 1/4. (h) g( t) = rect ( t) ∗ comb(2 t) This is the sum of multiple rectangle functions periodically repeated. 22. Plot these convolutions. (a) t + 1 t + 2 t g( t) = rect ∗ [δ ( t + 2) − δ ( t + 1)] = rect − rect 2 2 2 g(t) 1 -4 1 t -1 (b) g ( t ) = rect ( t ) ∗ tri ( t ) This is a challenging convolution because it is not so simple to do graphically (although you can get a rough idea of what it looks like that way) and it is tedious analytically. g (t ) = ∞ 1 2 ∫ rect (τ ) tri (t − τ ) dτ = ∫ tri (t − τ ) dτ −∞ − 1 2 Solutions 3-20 M. J. Roberts - 8/16/04 t < -3...
View Full Document

This note was uploaded on 06/19/2013 for the course ENSC 380 taught by Professor Atousa during the Spring '09 term at Simon Fraser.

Ask a homework question - tutors are online