1. EE104: Lecture 11 Outline Midterm Announcements Review of Last Lecture Sampling Nyquist Sampling Theorem Aliasing Signal Reconstruction via Interpolation Sampling in Frequency

2. Midterm Announcements No new homework this week Extra OHs next week: MT 11-12 Tomorrow’s OHs (Th 5:30) moved to Friday 11-12 Practice midterm l 10 bonus pts for taking any practice exam (1.5 hours) l Solutions will be available Monday Midterm Next Wednesday, 2/12, 12:50-2:05 in class Will cover through Friday’s lecture (Chapters 1-2) Open book and notes Review session Monday 2/10, 6:15-7:05, TCSEQ103 Wednesday review session cancelled

3. Review of Last Lecture Exponentials and Sinusoidal Functions Delta Function Train (Sampling Function) Fourier Transforms for Periodic Signals

4. Correction: Rects and Sincs.5T -.5T A t t.5B -.5B A/B f 1/B 2/B -1/B A f 1/T 2/T -1/T AT

5. Fourier Transforms for Periodic Signals x(t) t X(f) f 0 0 f 0 =1/T 0 x p (t)=x(t)   n  (t-nT 0 )=  n x(t)   (t-nT 0 ) X p (f)=X(f)  (1/T 0 )  n  (f-n/T 0 )=  n (1/T 0 ) X(n/T 0 )  (f-n/T 0 ) -2T 0 x(t)   (t+2T 0 ) T0T0 x(t)   (t-T 0 ) 2T 0 x(t)   (t-2T 0 ) x(t)   (t) -T 0 x(t)   (t-(-T 0 )) x p (t) =  n c n e j2  n/T 0 =  n c n  (f-n/T 0 ) C 0 =f 0 X(0) c 1 =f 0 X(f 0 ) c -1 f0f0 2f 0 -f 0 X p (f)

6. Sampling Sampling (Time): Sampling (Frequency) x s (t) 00 0 x(t) =  n  (t-nT s ) X s (f) 000 X(f) =  n  (t-n/T s ) * 1/T s -1/T s 1/T s

7. Nyquist Sampling Theorem A bandlimited signal [-B,B] is completely described by samples every T s <.5/B secs. Nyquist rate is 2B samples/sec Recreate signal from its samples by using a low pass filter in the frequency domain X s (f) X(f) B-B B X(f) 1/T s >2B -1/T s.5/T s

8. Aliasing Aliasing occurs when a signal is sampled below its Nyquist rate Repetitions in frequency domain overlap Distortion (aliasing) in frequency domain X s (f) X(f) B -B 1/T s <2B -1/T s B -B X´(f)

9. Signal Recovery and Interpolation Recover signal in frequency domain by passing sampled signal through LPF (rect) In time domain this becomes convolution of samples with sinc function Sinc function tracks signal changes between samples

10. Sampling in Frequency By duality, can recover time limited signal by sampling sufficiently fast in frequency Sampling in frequency is periodic repetition in time Recover time limited signal by windowing x(t) t X(f) f -T s 0 TsTs x s (t) 0 F s =1/T s X s (f).5T s -.5T s

11. Main Points Sampling in time is multiplication by delta train: transforms to convolution with delta train Signals that are not sampled faster than the Nyquist rate have aliasing or distortion: can’t be recovered from samples. Recover signal from samples using sinc interpolation in time: low pass filtering in frequency Sampling in the frequency domain becomes repetition in time.