1. EE104: Lecture 9 Outline Announcements HW 2 due today, HW 3 posted Midterm scheduled for 2/12, may move to 2/14. Review of Last Lecture Signal Bandwidth Dirac Delta Function and its Properties Filter Impulse and Frequency Response Exponentials and Sinusoidal Functions Delta Function Train (Sampling Function)

2. z(2-  ) 2 Review of Last Lecture: Convolution y(t)=x(t) * z(t)=  x(  )z(t-  )d  Flip one signal and drag it across the other Area under product at drag offset t is y(t). tt+1t-1 z(t-  ) 01 x(t) 01 z(t) t t 01 x(  )   -6 01 y(t) -2 2 0 2 t z(-2-  ) z(-1.99-  ).01 z(0-  ) 2 z(1-  ) 2 -6 -4 z(-1-  ) 1 z(  )  x(  ) 

3. Convolution Tips and Properties Tips for plotting convolution: Plot all zero points Plot all maxs and mins Plot any other transition points Find any straight lines connecting points Properties x * y=y * x x * (y+z)=x * y+x * z (x * y) * z=x * (y * z)

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

5. Dirac Delta Function Defined by two equations  (t)=0, t  0   (t)dt=1 Alternatively defined as a limit  (t)=lim  0 (1/  )rect(t/  ) 0  (t) 0   

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

7. 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(f 0 ) h(t)  (t)y(t)=h(t) *  (t)=h(t) H(f) LTI Filter Y(f)=H(f)1=H(f)  y(t)=H(f 0 ) e j2  f 0 t e j2  f 0 t

8. Sinusoids and Exponentials Exponentials become a shifted delta Sinusoids become two shifted deltas The Fourier Transform of a periodic signal is a weighted train of deltas fcfc  (f-f c ) Ae j2  f c t  2Acos(2  f c t)  fcfc  (f-f c ) -f c  (f+f c ) H(f)  (f-f c ) H(f c )  (f-f c ) H(f c ) e j2  f c t

9. Delta Function Trains (Sampling Function) T s  n  (t-nT s ) 0  n  (t-n/T s ) 0TsTs 2T s 3T s -T s -2T s -3T s 1/T s -1/T s The key Fourier transforms we will use are rectangles, sincs, exponentials and sinusoids, deltas and delta function trains

10. Main Points 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. Exponentials and sinusoids in time are simple combinations of delta functions in frequency Delta function trains in time is a delta function train in frequency Only REALLY need to know a few FT pairs and properties