READING ASSIGNMENTS. Signal Processing First. SYSTEMS Process Signals LECTURE OBJECTIVES. This Lecture: Lecture 8 Sampling & Aliasing.
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1 Signal Proceing Firt Lecture 8 Sampling & Aliaing READING ASSIGNMENTS Thi Lecture: Chap 4, Section 4- and 4-2 Replace Ch 4 in DSP Firt, pp Other Reading: Recitation: Strobe Demo (Sect 4-3 Next Lecture: Chap. 4 Sect. 4-4 and 4-5 9/4/ , JH McClellan & RW Schaer 9/4/ , JH McClellan & RW Schaer 3 LECTURE OBJECTIVES SYSTEMS Proce Signal SAMPLING can caue ALIASING Sampling Theorem Sampling Rate > 2(Highet Frequency Spectrum or digital ignal, x[n] Normalized Frequency = ωt = + 9/4/ , JH McClellan & RW Schaer 4 ALIASING x(t SYSTEM PROCESSING GOALS: Change x(t into y(t For example, more BASS y(t Improve x(t, e.g., image deblurring Extract Inormation rom x(t 9/4/ , JH McClellan & RW Schaer 5
2 Sytem IMPLEMENTATION ANALOG/ELECTRONIC: Circuit: reitor, capacitor, op-amp x(t ELECTRONICS y(t DIGITAL/MICROPROCESSOR Convert x(t to number tored in memory SAMPLING x(t SAMPLING PROCESS Convert x(t to number x[n] n i an integer; x[n] i a equence o value Think o n a the torage addre in memory UNIFORM SAMPLING at t = nt IDEAL: x[n] = x(nt x(t A-to-D x[n] COMPUTER y[n] D-to-A y(t x(t C-to-D x[n] 9/4/ , JH McClellan & RW Schaer 6 9/4/ , JH McClellan & RW Schaer 7 = 00Hz SAMPLING RATE, SAMPLING RATE ( =/T NUMBER o SAMPLES PER SECOND T = 25 microec = 8000 ample/ec UNITS ARE HERTZ: 8000 Hz UNIFORM SAMPLING at t = nt = n/ IDEAL: x[n] = x(nt =x(n/ x(t x[n]=x(nt C-to-D = 2 khz = 500Hz 9/4/ , JH McClellan & RW Schaer 8 9/4/ , JH McClellan & RW Schaer 9
3 SAMPLING THEOREM HOW OFTEN? DEPENDS on FREQUENCY o SINUSOID ANSWERED by SHANNON/NYQUIST Theorem ALSO DEPENDS on RECONSTRUCTION Recontruction? Which One? Given the ample, draw a inuoid through the value = co(0.4π n When n i an integer co(0.4π n = co(2.4π n 9/4/ , JH McClellan & RW Schaer 0 9/4/ , JH McClellan & RW Schaer STORING DIGITAL SOUND x[n] i a SAMPLED SINUSOID A lit o number tored in memory EXAMPLE: audio CD CD rate i 44,00 ample per econd 6-bit ample Stereo ue 2 channel Number o byte or minute i (6/8 X 60 X 4400 = Mbyte 9/4/ , JH McClellan & RW Schaer 2 DISCRETE-TIME SINUSOID Change x(t into x[n] DERIVATION x( t = Aco( ωt + ϕ x[ n] = x( nt = Aco( ω x[ n] = Aco( n + ϕ ˆ ω ω = ω = T nt = Aco(( ω T n + ϕ + ϕ 9/4/ , JH McClellan & RW Schaer 3 DEFINE DIGITAL FREQUENCY
4 DIGITAL FREQUENCY ωˆ SPECTRUM (DIGITAL ωˆ VARIES rom 0 to, a varie rom 0 to the ampling requency UNITS are radian, not rad/ec DIGITAL FREQUENCY i NORMALIZED = = khz * 0. (0. = Aco( (00( n /000 + ϕ = ωt = 9/4/ , JH McClellan & RW Schaer 4 9/4/ , JH McClellan & RW Schaer 5 SPECTRUM (DIGITAL??? The REST o the STORY = =00 Hz *? ( = Aco( (00( n /00 + ϕ Spectrum o x[n] ha more than one line or each complex exponential Called ALIASING MANY SPECTRAL LINES x[n] i zero requency??? 9/4/ , JH McClellan & RW Schaer 6 SPECTRUM i PERIODIC with period = Becaue A co( n + ϕ = A co(( + n +ϕ 9/4/ , JH McClellan & RW Schaer 7
5 ALIASING DERIVATION ω Other Frequencie give the ame ˆ x ( t = co(400π t ampled at = 000 Hz n x n] = co(400π = co(0.4π [ n 000 x2 ( t = co(2400π t ampled at = 000 Hz n x n] = co(2400π = co(2.4π 2[ n 000 x [ n] = co(2.4π n = co(0.4π n + n = co(0.4π 2 n x n] = x [ ] 2400π 400π = (000 2[ n 9/4/ , JH McClellan & RW Schaer 8 ALIASING DERIVATION 2 Other Frequencie give the ame I x(t = A co( ( + t + ϕ ˆ ( + then : ω = = + = ωt = + 9/4/ , JH McClellan & RW Schaer 9 and we want : x[n] = Aco( n +ϕ t n ALIASING CONCLUSIONS ADDING or 2 or to the FREQ o x(t give exactly the ame x[n] The ample, x[n] = x(n/ are EXACTLY THE SAME VALUES GIVEN x[n], WE CAN T DISTINGUISH o FROM ( o + or ( o + 2 NORMALIZED FREQUENCY DIGITAL FREQUENCY = ωt = + 9/4/ , JH McClellan & RW Schaer 20 9/4/ , JH McClellan & RW Schaer 2
![SPECTRUM or x[n] SPECTRUM (MORE LINES PLOT veru NORMALIZED FREQUENCY INCLUDE ALL SPECTRUM LINES ALIASES ADD MULTIPLES o SUBTRACT MULTIPLES o FOLDED ALIASES (to be dicued later ALIASES o NEGATIVE](/docs-images/88/115521670/images/6-2.jpg "FREQS = = khz.8π * 0. (0. = Aco( (00( n /000 + ϕ *.8π 9/4/2003 2003, JH McClellan & RW Schaer 22 9/4/2003 2003, JH McClellan & RW Schaer 23 SPECTRUM (ALIASING CASE SAMPLING GUI (con2di = = 80kHz 2.5π.")
6 SPECTRUM or x[n] SPECTRUM (MORE LINES PLOT veru NORMALIZED FREQUENCY INCLUDE ALL SPECTRUM LINES ALIASES ADD MULTIPLES o SUBTRACT MULTIPLES o FOLDED ALIASES (to be dicued later ALIASES o NEGATIVE FREQS = = khz.8π * 0. (0. = Aco( (00( n /000 + ϕ *.8π 9/4/ , JH McClellan & RW Schaer 22 9/4/ , JH McClellan & RW Schaer 23 SPECTRUM (ALIASING CASE SAMPLING GUI (con2di = = 80kHz 2.5π.5π * 0.5π * * 0.5π.5π 2.5π = Aco( (00( n /80 + ϕ 9/4/ , JH McClellan & RW Schaer 24 9/4/ , JH McClellan & RW Schaer 25
7 SPECTRUM (FOLDING CASE = = 25Hz *.6π * 0.4π 0.4π.6π x[ n] = Aco( (00( n /25 + ϕ 9/4/ , JH McClellan & RW Schaer 26
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