1. Continuous-Time Convolution EE 313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin

2. 4 - 2 Convolution Integral Commonly used in engineering, science, math Convolution properties –Commutative: f 1 (t) * f 2 (t) = f 2 (t) * f 1 (t) –Distributive: f 1 (t) * [f 2 (t) + f 3 (t)] = f 1 (t) * f 2 (t) + f 1 (t) * f 3 (t) –Associative: f 1 (t) * [f 2 (t) * f 3 (t)] = [f 1 (t) * f 2 (t)] * f 3 (t) –Shift: If f 1 (t) * f 2 (t) = c(t), then f 1 (t) * f 2 (t - T) = f 1 (t - T) * f 2 (t) = c(t - T). –Convolution with impulse, f(t) *  (t) = f(t) –Convolution with shifted impulse, f(t) *  (t-T) = f(t-T) important later in modulation

3. 4 - 3 Graphical Convolution Methods From the convolution integral, convolution is equivalent to –Rotating one of the functions about the y axis –Shifting it by t –Multiplying this flipped, shifted function with the other function –Calculating the area under this product –Assigning this value to f 1 (t) * f 2 (t) at t

4. 4 - 4 3  2 f()f() 2 -2 + t2 + t g(t-  ) * 2 2 t f(t)f(t) -22 3 t g(t)g(t) Graphical Convolution Example Convolve the following two functions: Replace t with  in f(t) and g(t) Choose to flip and slide g(  ) since it is simpler and symmetric Functions overlap like this:

5. 4 - 5 3  2 f()f() 2 -2 + t2 + t g(t-  ) 3  2 f()f() 2 -2 + t2 + t g(t-  ) Graphical Convolution Example Convolution can be divided into 5 parts I. t < -2 Two functions do not overlap Area under the product of the functions is zero II. -2  t < 0 Part of g(t) overlaps part of f(t) Area under the product of the functions is

6. 4 - 6 Graphical Convolution Example III. 0  t < 2 Here, g(t) completely overlaps f(t) Area under the product is just IV. 2  t < 4 Part of g(t) and f(t) overlap Calculated similarly to -2  t < 0 V. t  4 g(t) and f(t) do not overlap Area under their product is zero 3  2 f()f() 2 -2 + t2 + t g(t-  ) 3  2 f()f() 2 -2 + t2 + t g(t-  )

7. 4 - 7 Graphical Convolution Example Result of convolution (5 intervals of interest): t y(t)y(t) 024-2 6

8. 4 - 8 Convolution Demos Johns Hopkins University Demonstrations http://www.jhu.edu/~signals Convolution applet to animate convolution of simple signals and hand-sketched signals Convolve two rectangular pulses of same width gives a triangle (see handout E) Some conclusions from the animations Convolution of two causal signals gives a causal result Non-zero duration (called extent) of convolution is the sum of extents of the two signals being convolved

9. 4 - 9 Transmit One Bit Transmission over communication channel (e.g. telephone line) is analog hh t 1 pp t A ‘1’ bit t pp -A-A ‘0’ bit Model channel as LTI system with impulse response h(t) Communication Channel inputoutput x(t)x(t)y(t)y(t) t t receive ‘1’ bit -A T h receive ‘0’ bit h+ph+p t h+ph+p hh hh Assume that T h < T p A T h

10. 4 - 10 Transmit Two Bits (Interference) Transmitting two bits (pulses) back-to-back will cause overlap (interference) at the receiver How do we prevent intersymbol interference at the receiver? hh t 1 Assume that T h < T p t pp A ‘1’ bit ‘0’ bit pp *= -A T h t pp ‘1’ bit ‘0’ bit h+ph+p intersymbol interference

11. 4 - 11 Transmit Two Bits (No Interference) Prevent intersymbol interference by waiting T h seconds between pulses (called a guard period) Disadvantages? hh t 1 Assume that T h < T p *= t pp A ‘1’ bit ‘0’ bit h+ph+p t -A T h pp ‘1’ bit ‘0’ bit h+ph+p hh

12. 4 - 12 h[n]h[n] y[n]y[n]x[n]x[n] LTI system represented by its impulse response h(t)h(t) y(t)y(t)x(t)x(t) Discrete-time Convolution Preview Discrete-time convolution For every value of n, we compute a new summation Continuous-time convolution For every value of t, we compute a new integral

13. 4 - 13 z -1 … … x[n]x[n]  y[n]y[n] h[0]h[1]h[2]h[N-1] Discrete-time Convolution Preview Assuming that h[n] has finite duration from n = 0, …, N-1 Block diagram of an implementation (finite impulse response digital filter): see slide 2-4

14. 4 - 14 Philosophy Pillars of electrical engineering (related) Fourier analysis Probability and random processes Pillars of computer engineering (related) System state Complexity Finite-state machines for digital input/output Finite number of states Models all possible input- output combinations Can two outputs be true at the same time? Given output observation, work backwards to inputs to see if output is possible This is called observability EE 313 is pre-requisite for EE 351K

15. 4 - 15 Corporate Technical Ladder Test Engineer BS degree Test other people’s designs Starting salary: $65,000 Design Engineer MS degree, or BS degree plus 2 years experience and design short courses Design new products Starting salary: $75,000 What about the Ph.D.? ¾ of Ph.D.’s to industry ¼ of Ph.D.’s to academia BSEE Tech. BSEE MSEE PhDEE 1 PhDEE 2 Technician Test Eng. Design Eng. Proj. Management Technical Staff (R&D) VP, Eng. CTO Director Eng. (1)Ph.D. based on system prototyping (2)Ph.D. with significant theoretical results