Signal Processing First Lecture 8 Sampling Aliasing 1142020

Signal Processing First Lecture 8 Sampling & Aliasing 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 1

READING ASSIGNMENTS § This Lecture: § Chap 4, Sections 4 -1 and 4 -2 § Replaces Ch 4 in DSP First, pp. 83 -94 § Other Reading: § Recitation: Strobe Demo (Sect 4 -3) § Next Lecture: Chap. 4 Sects. 4 -4 and 4 -5 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 2

LECTURE OBJECTIVES § SAMPLING can cause ALIASING § Nyquist/Shannon Sampling Theorem § Sampling Rate (fs) > 2 fmax(Signal bandwidth) § Spectrum for digital signals, x[n] § Normalized Frequency ALIASING 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 3

SYSTEMS Process Signals x(t) SYSTEM y(t) § PROCESSING GOALS: § We need to change x(t) into y(t) for many engineering applications: § For example, more BASS, image deblurring, denoising, etc 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 4

System IMPLEMENTATION § ANALOG/ELECTRONIC: § Circuits: resistors, capacitors, op-amps x(t) y(t) ELECTRONICS § DIGITAL/MICROPROCESSOR § Convert x(t) to numbers stored in memory x(t) 11/4/2020 A-to-D x[n] COMPUTER y[n] © 2003, JH Mc. Clellan & RW Schafer D-to-A y(t) 5

SAMPLING x(t) § SAMPLING PROCESS § Convert x(t) to numbers x[n] § “n” is an integer; x[n] is a sequence of values § Think of “n” as the storage address in memory § UNIFORM SAMPLING at t = n. Ts § IDEAL: x[n] = x(n. Ts) x(t) 11/4/2020 A-to-D x[n] © 2003, JH Mc. Clellan & RW Schafer 6

SAMPLING RATE, fs § SAMPLING RATE (fs) § fs =1/Ts § NUMBER of SAMPLES PER SECOND § Ts = 125 microsec fs = 8000 samples/sec • UNITS ARE HERTZ: 8000 Hz § UNIFORM SAMPLING at t = n. Ts = n/fs § IDEAL: x[n] = x(n. Ts)=x(n/fs) x(t) 11/4/2020 A-to-D x[n]=x(n. Ts) © 2003, JH Mc. Clellan & RW Schafer 7

11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 8

SAMPLING THEOREM § HOW OFTEN ? § DEPENDS on FREQUENCY of SINUSOID § ANSWERED by NYQUIST/SHANNON Theorem § ALSO DEPENDS on “RECONSTRUCTION” 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 9

Reconstruction? Which One? Given the samples, draw a sinusoid through the values 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 10

STORING DIGITAL SOUND § x[n] is a SAMPLED SINUSOID § A list of numbers stored in memory § EXAMPLE: audio CD § CD rate is 44, 100 samples per second § 16 -bit samples § Stereo uses 2 channels § Number of bytes for 1 minute is § 2 X (16/8) X 60 X 44100 = 10. 584 Mbytes 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 11

DISCRETE-TIME SINUSOID § Change x(t) into x[n] DERIVATION DEFINE DIGITAL FREQUENCY 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 12

DIGITAL FREQUENCY § VARIES from 0 to 2 p, as f varies from 0 to the sampling frequency § UNITS are radians, not rad/sec § DIGITAL FREQUENCY is NORMALIZED 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 13

SPECTRUM (DIGITAL) – 2 p(0. 1) 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 2 p(0. 1) 14

SPECTRUM (DIGITAL) ? ? – 2 p (1) 2 p(1) x[n] is zero frequency? ? ? 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 15

The REST of the STORY § Spectrum of x[n] has more than one line for each complex exponential § Called ALIASING § MANY SPECTRAL LINES § SPECTRUM is PERIODIC with period = 2 p § Because 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 16

ALIASING DERIVATION § Other Frequencies give the same 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 17

ALIASING DERIVATION– 2 § Other Frequencies give the same 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 18

ALIASING CONCLUSIONS § ADDING fs or 2 fs or –fs to the FREQ of x(t) gives exactly the same x[n] § The samples, x[n] = x(n/ fs ) are EXACTLY THE SAME VALUES § GIVEN x[n], WE CAN’T DISTINGUISH fo FROM (fo + fs ) or (fo + 2 fs ) 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 19

NORMALIZED FREQUENCY § DIGITAL FREQUENCY 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 20

SPECTRUM for x[n] § PLOT versus NORMALIZED FREQUENCY § INCLUDE ALL SPECTRUM LINES § ALIASES § ADD MULTIPLES of 2 p § SUBTRACT MULTIPLES of 2 p § FOLDED ALIASES § (to be discussed later) § ALIASES of NEGATIVE FREQS 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 21

SPECTRUM (MORE LINES) – 1. 8 p 11/4/2020 – 0. 2 p 2 p(0. 1) © 2003, JH Mc. Clellan & RW Schafer 1. 8 p 22

SPECTRUM (ALIASING CASE) – 2. 5 p 11/4/2020 – 1. 5 p – 0. 5 p © 2003, JH Mc. Clellan & RW Schafer 1. 5 p 23

SAMPLING GUI (con 2 dis) 11/4/2020 © 2003, JH Mc. Clellan & RW Schafer 24

SPECTRUM (FOLDING CASE) – 1. 6 p 11/4/2020 – 0. 4 p © 2003, JH Mc. Clellan & RW Schafer 1. 6 p 25