EE104: Lecture 8 Outline

1. EE104: Lecture 8 Outline • Review of Last Lecture • Convolution Review • Signal Bandwidth • Dirac Delta Function and its Properties • Filter Impulse and Frequency Response

2. Review of Last Lecture • Time Scaling • Duality • Frequency Shifting (Modulation) • Multiplication  Convolution • Convolution  Multiplication • Filter analysis often easier in frequency domain LTI Filter x(t) y(t)=h(t)*x(t) h(t) H(f) X(f) Y(f)=H(f)X(f)

3. 2 2 z(-4-t) z(-6-t) z(1-t) z(-2-t) z(-1.99-t) z(0-t) z(2-t) z(-1-t) 2 1 .01 Convolution Review • y(t)=x(t)*z(t)= x(t)z(t-t)dt • Flip one signal and drag it across the other • Area under product at drag offset t is y(t). z(t) x(t) z(t-t) x(t) z(t) t t t t t t-1 t t+1 -1 0 1 -1 0 1 x(t) t 2 -6 -1 0 1 -4 -2 y(t) 2 -4 t 0 -2 -6 -1 0 1

4. Signal Bandwidth • For bandlimited signals, bandwidth B defined as range of positivefrequencies for which |X(f)|>0. • In practice, all signals time-limited • Not bandlimited • Need alternate bandwidth definition 3dB Bandlimited Null-to-Null |X(f)| |X(f)| -3dB |X(f)| 0 0 0 2B 2B 2B

5. Dirac Delta Function • Defined by two equations • d(t)=0, t=0 • d(t)dt=1 • Alternatively defined as a limit • d(t)=limt0 (1/t)rect(t/t) d(t) 0 d(t) 0

6. Delta Function Properties • x(t)*d(t)=x(t) • d(t)1 • DC signals are d functions in frequency.

7. d(t) y(t)=h(t)*d(t)=h(t) ej2pf0t 1 Y(f)=H(f0) ej2pf0t Y(f)=H(f)1=H(f) Filter Response • Impulse Response (Time Domain) • Filter output in response to a delta input • Frequency Response (Freq. Domain) • Fourier transform of impulse response • The response of a filter to an exponential input the same exponential weighted by H(f0) LTI Filter h(t) H(f)

8. Main Points • Convolution is a drag (and a flip) • Signal bandwidth definition depends on its use • Dirac delta function is a mathematical construct that is useful in analyzing signals and filters • Filter impulse response defined as filter output to delta input • Filter frequency response is Fourier transform of its impulse response