Reminder. ECE 2026 Summer Quiz 1 Review Session Tonight. Quiz 1 Coverage

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1 ECE 2026 Summer 208 LECTURE #8 More Sampling & Aliaing June, 208 Reminder HW4 due tomorrow Lab4 Thurday i eay Quiz i Wedneday Cloed book, except or one 2-ided heet o handwitten note Calculator allowed No martphone/tablet/laptop/wifi/etc Quiz Coverage Chapter through 4 (but not Sect 4-3) Lecture through 7 (plu any review in 8) HW # through 4 Lab 0 through 3 (MATLAB air game) Quiz Review Seion Tonight 6:30-8pm Clough Room 423 Optional Come with quetion 6//208 4

2 SAMPLING SINUSOIDS SAMPLING THEOREM How at mut we ample? t = n/ REDUCE 2 n = t 3 Equivalent condition when ampling inuoid: more than 2 ample per cycle. Ueul Fact: Sampling Hz too Slow Contruct a Chirp that ramp up rom khz to 9kHZ in 4 econd: x( = co(2 0 t ) y(=? 3 khz co(2 ( 0 )t ) = 2 0 khz 0 4 t 0 /2 (Hz) = 8000; t=0:/:4; x = co(2000*pi*t *pi*t.^2); oundc(x,); pectrogram(x,2^9,[],2^4,,'yaxi');? 2

3 Pop Quiz Recontruction? Which One? Given the ample, draw a inuoid through the value t=0:/:4; x = co(a*pi*t - B*pi*in(C*pi*); pectrogram(x,2^9,[],2^4,,'yaxi'); = A = B = C = co(0.4 n) When n i an integer co(0.4 n) co(2.4 n) Occam razor -> pick lowet requency inuoid Recontruction or Sinuoid SPECTRUM (DIGITAL). Reduce digital requency to principal range 2. Subtitute n = t Key reult: recontructed requency will alway be < /2! ˆ 2 khz * A co( 2 ( 00)( n / 000) ) Max requency = /2 6//208 3

4 SPECTRUM (DIGITAL)??? ˆ 2 00 Hz * 2 x[n] i zero requency???? 2 Aco(2 (00)( n /00) ) ˆ The REST o the STORY Spectrum o x[n] ha more than one line or each complex exponential Called ALIASING MANY SPECTRAL LINES SPECTRUM i PERIODIC with period = 2 Becaue A co ˆ n A co ˆ 2 n SPECTRUM or x[n] SPECTRUM (MORE LINES) PLOT veru NORMALIZED FREQUENCY INCLUDE ALL SPECTRUM LINES ALIASES ADD MULTIPLES o 2 SUBTRACT MULTIPLES o 2 FOLDED ALIASES (to be dicued later) ALIASES o NEGATIVE FREQS ˆ 2 khz.8 * Aco(2 (00)( n /000) ) *.8 6// , JH McClellan & RW Schaer 6 4

5 SPECTRUM (ALIASING CASE) SAMPLING GUI (con2di) ˆ kHz * * * Aco(2 (00)( n /80) ) 6// , JH McClellan & RW Schaer 7 SPECTRUM (FOLDING CASE) Aliaing Demo with Chirp 2 25Hz *.6 * Aco(2 (00)( n /25) ) t = 0:/8000:4; xx=co(2*pi*000*t.*( + ); plotpec(xx + j*e-9,8000) grid on, hg oundc(xx,8000) = 8000 Hz 6// , JH McClellan & RW Schaer 9 6//208 ECE-2025 Spring-20 jmc 20 5

6 SAMPLING GUI (con2di) Terminology Nyquit Rate = 2 max A ignal i bandlimited i it ha a (inite) maximum requency max Pop Quiz: I a quare wave bandlimited? 6//208 ECE-2025 Spring-20 jmc 2 6// , JH McClellan & RW Schaer 22 SPECTRUM or x[n] INCLUDE ALL SPECTRUM LINES ALIASES ADD INTEGER MULTIPLES o 2 and 2 FOLDED ALIASES ALIASES o NEGATIVE FREQS PLOT veru NORMALIZED FREQUENCY i.e., DIVIDE o by ˆ ( 0) 2 2 6// , JH McClellan & RW Schaer 23 EXAMPLE: SPECTRUM x[n] = Aco(0.2 n+ ) 0.2 and 0.2 ALIASES: {2.2, 4.2, 6.2, } & {-.8,-3.8, } EX: x[n] = Aco(4.2 n+ ) ALIASES o NEGATIVE FREQ: {.8,3.8,5.8, } & {-2.2, -4.2 } 6// , JH McClellan & RW Schaer 24 6

7 SPECTRUM (MORE LINES) SPECTRUM (ALIASING CASE) ˆ 2 khz.8 * Aco(2 (00)( n /000) ) *.8 ˆ 2 80Hz * * * Aco(2 (00)( n /80) ) Principal alia : ˆ.(000) 00 Hz 2 x( Aco(2 00t ) ˆ Principal alia : ˆ.25(80) 20 Hz 2 x( Aco(2 20t ) Principal alia i alway between ˆ 6//208 Principal alia 2003, ijh alway McClellan & RW between Schaer 25 6// , JH McClellan & RW Schaer 26 ˆ DIGITAL FREQ AGAIN SPECTRUM (FOLDING CASE) ˆ T ˆ T 2 ALIASING FOLDED ALIAS 6// , JH McClellan & RW Schaer Hz *.6 * Aco(2 (00)( n /25) ) Principal alia (with olding) : ˆ.2(25) 25Hz 2 x( Aco(2 25t ) Principal alia i alway between ˆ 6// , JH McClellan & RW Schaer 28 7

8 SPECTRUM Explanation o SAMPLING THEOREM How do we prevent aliaing? Guarantee original ignal i principal alia: ˆ ˆ 0 0 * 0 2 ˆ // , JH McClellan & RW Schaer ˆ 0 0 D-to-A Recontruction x( A-to-D x[n] COMPUTER y[n] Create continuou y( rom y[n] IDEAL D-to-A: I you have ormula or y[n] Invert ampling (t=nt ) by n= t D-to-A y( y[n] = Aco(0.2 n+ ) with = 8000 Hz y( = Aco(0.2 (8000+ ) = Aco(2 (800)t+ ) 6// , JH McClellan & RW Schaer 30 FREQUENCY DOMAINS D-to-A i AMBIGUOUS! x( A-to-D x[n] ( ) ˆ 2 2 y[n] ˆ ˆ D-to-A y( ˆ 2 ALIASING Given y[n], which y( do we pick??? INFINITE NUMBER o y( PASSING THRU THE SAMPLES, y[n] D-to-A RECONSTRUCTION MUST CHOOSE ONE OUTPUT RECONSTRUCT THE SMOOTHEST ONE THE LOWEST FREQ, i y[n] = inuoid 6// , JH McClellan & RW Schaer 3 6// , JH McClellan & RW Schaer 32 8

9 SPECTRUM (ALIASING CASE) 2 * * * 80Hz Aco(2 (00)( n /80) ) DEMOS rom CHAPTER 4 CD-ROM DEMOS SAMPLING DEMO (con2di GUI) Dierent Sampling Rate Aliaing o a Sinuoid STROBE DEMO Synthetic v. Real Televiion SAMPLES at 30 p Sampling & Recontruction 6// , JH McClellan & RW Schaer 33 6// , JH McClellan & RW Schaer 34 FOLDING DIAGRAM 2000 Hz Recontruction (D-to-A) CONVERT STREAM o NUMBERS to x( CONNECT THE DOTS INTERPOLATION y[k] INTUITIVE, convey the idea y( kt (k+)t t 6// , JH McClellan & RW Schaer 35 6// , JH McClellan & RW Schaer 36 9

10 SAMPLE & HOLD DEVICE SQUARE PULSE CASE CONVERT y[n] to y( y[k] hould be the value o y( at t = kt Make y( equal to y[k] or kt -0.5T < t < kt +0.5T y( y[k] STAIR-STEP APPROXIMATION kt (k+)t t 6// , JH McClellan & RW Schaer 37 6// , JH McClellan & RW Schaer 38 OVER-SAMPLING CASE MATH MODEL or D-to-A EASIER TO RECONSTRUCT SQUARE PULSE: 6// , JH McClellan & RW Schaer 39 6// , JH McClellan & RW Schaer 40 0

11 EXPAND the SUMMATION n y[ n] t nt ) y[0] y[] t T ) y[2] t 2T ) SUM o SHIFTED PULSES t-nt ) WEIGHTED by y[n] CENTERED at t=nt SPACED by T RESTORES REAL TIME 6// , JH McClellan & RW Schaer 4 6// , JH McClellan & RW Schaer 42 TRIANGULAR PULSE (2X) OPTIMAL PULSE? CALLED BANDLIMITED INTERPOLATION 6// , JH McClellan & RW Schaer 43 in t T t T 0 or t or t T, 2T, 6// , JH McClellan & RW Schaer 44

12 Recontruct with Ideal (2x) SAMPLING SINUSOIDS t = n/ REDUCE 2 n = t 3 6// , JH McClellan & RW Schaer 45 SAMPLING THEOREM How at mut we ample? Aliaing Demo with Chirp t = 0:/:4; xx=co(2*pi*000*t.*( + ); plotpec(xx + j*e-9,8000) grid on, hg oundc(xx,) =? 6//208 ECE-2025 Spring-20 jmc 48 2

13 Terminology Input i a Lit o Number? Nyquit Rate = 2 max A ignal i bandlimited i it ha a maximum requency max? Pop Quiz: I a quare wave bandlimited? 6// , JH McClellan & RW Schaer 49 Recontruction (D-to-A) SAMPLE & HOLD DEVICE Convert lit o number to waveorm Connect the dot, ill in the gap interpolate y( y[k] kt (k+)t t INTUITIVE, convey the idea CONVERT y[n] to y( y[k] hould be the value o y( at t = kt Make y( equal to y[k] or kt -0.5T < t < kt +0.5T y( y[k] kt (k+)t t STAIR-STEP APPROXIMATION 6// , JH McClellan & RW Schaer 5 6// , JH McClellan & RW Schaer 52 3

14 SQUARE PULSE CASE OVER-SAMPLING CASE EASIER TO RECONSTRUCT 6// , JH McClellan & RW Schaer 53 6// , JH McClellan & RW Schaer 54 MATH MODEL or D-to-A SQUARE PULSE: n EXPAND the SUMMATION y[ n] t nt ) y[0] y[] t T ) y[2] t 2T ) SUM o SHIFTED PULSES t-nt ) WEIGHTED by y[n] CENTERED at t=nt SPACED by T RESTORES REAL TIME 6// , JH McClellan & RW Schaer 55 6// , JH McClellan & RW Schaer 56 4

15 TRIANGULAR PULSE (2X) 6// , JH McClellan & RW Schaer 57 6// , JH McClellan & RW Schaer 58 OPTIMAL PULSE? Recontruct with Ideal (2x) CALLED BANDLIMITED INTERPOLATION in t T t T 0 or t or t T, 2T, 6// , JH McClellan & RW Schaer 59 6// , JH McClellan & RW Schaer 60 5