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Prof. Nizamettin AYDIN

naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin

Digital Signal Processing

2

Lecture 12

Frequency Response Frequency Response

of FIR Filtersof FIR Filters

Digital Signal Processing

3

License Info for SPFirst Slides

• This work released under a Creative Commons Licensewith the following terms: • Attribution • The licensor permits others to copy, distribute, display, and perform the work. In return, licensees must give the original authors credit. • Non-Commercial • The licensor permits others to copy, distribute, display, and perform the work. In return, licensees may not use the work for commercial purposes-unless they get the licensor"s permission. • Share Alike • The licensor permits others to distribute derivative works only under a license identical to the one that governs the licensor"s work. •Full Text of the License •This (hidden) page should be kept with the presentation 4

READING ASSIGNMENTS

• This Lecture: - Chapter 6, Sections 6-1, 6-2, 6-3, 6-4, & 6-5 • Other Reading: - Recitation: Chapter 6 • FREQUENCY RESPONSE EXAMPLES - Next Lecture: Chap. 6, Sects. 6-6, 6-7 & 6-8 5

LECTURE OBJECTIVES

• SINUSOIDALINPUT SIGNAL - DETERMINE the FIR FILTER OUTPUT • FREQUENCY RESPONSE of FIR - PLOTTING vs. Frequency - MAGNITUDE vs. Freq - PHASE vs. Freq wwwjeHjjjeeHeHÐ= MAG PHASE 6

DOMAINS: Time & Frequency

•Time-Domain: "n" = time -x[n] discrete-time signal -x(t) continuous-time signal •Frequency Domain (sum of sinusoids) - Spectrum vs. f (Hz) - ANALOG vs. DIGITAL - Spectrum vs. omega-hat • Move back and forth QUICKLY 2 7

LTI SYSTEMS

• LTI:

Linear & Time-Invariant

• COMPLETELY CHARACTERIZED by: -IMPULSE RESPONSEh[n] -CONVOLUTION : y[n] = x[n]*h[n] • The "rule"defining the system can ALWAYS be re- written as convolution • FIR Example: h[n] is same as bk 8

POP QUIZ

• FIR Filter is "FIRST DIFFERENCE" -y[n] = x[n] -x[n-1] • Write output as a convolution - Need impulse response - Then, another way to compute the output: ]1[][][--=nnnhdd ()][]1[][][nxnnny*--=dd 9

DIGITAL "FILTERING"

• CONCENTRATE on the

SPECTRUMSPECTRUM

• SINUSOIDAL INPUT - INPUT x[n] = SUM of SINUSOIDS - Then, OUTPUT y[n] = SUM of SINUSOIDS

FILTERD-to-AA-to-Dx(t)y(t)y[n]x[n]

wˆwˆ 10

FILTERING EXAMPLE

• 7-point AVERAGER - Removes cosine • By making its amplitude (A) smaller • 3-point AVERAGER - Changes A slightly 6 0 717
k knxny 2 0 313
k knxny 11

3-pt AVG EXAMPLE

USE PAST VALUES

400for)4/8/2cos()02.1(][ :Input££++=nnnxnpp

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7-pt FIR EXAMPLE (AVG)

CAUSAL: Use Previous

LONGER OUTPUT

400for)4/8/2cos()02.1(][ :Input££++=nnnxnpp

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SINUSOIDAL RESPONSE

• INPUT: x[n] = SINUSOID • OUTPUT: y[n] will also be a SINUSOID - Different Amplitude and Phase -SAMEFrequency • AMPLITUDE & PHASE CHANGE - Called the FREQUENCY RESPONSEFREQUENCY RESPONSE 14

DCONVDEMO: MATLAB GUI

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COMPLEX EXPONENTIAL

M kM k k knxkhknxbny 00

FIR DIFFERENCE EQUATION

¥<<¥-=neAenxnjjwjˆ][

x[n] is the input signal-a complex exponential 16

COMPLEX EXP OUTPUT

• Use the FIR "Difference Equation" njjeAeHwjwˆ)ˆ(= M kknjj kM k k eAebknxbny

0)(ˆ

0 ][][wj njjM kkj keAeebwjwˆ 0)( 17

FREQUENCY RESPONSE

• Complex-valued formula - Has MAGNITUDE vs. frequency - And PHASE vs. frequency • Notation:

FREQUENCY

RESPONSE

• At each frequency, we can DEFINE )ˆ(of place in )(ˆwwHeHj kjM k kebHwwˆ

0)ˆ(-

=∑=kjM k kjebeHwwˆ

0ˆ)(-

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EXAMPLE 6.1

EXPLOIT

SYMMETRY

}1,2,1{}{=kb )ˆcos22()2(21)(ˆˆˆˆˆ

2ˆˆwwwwwwww+=++=++=-----

j jjjjjjeeeeeeeH ww w wwˆ)( is Phaseand)

ˆcos22()( is Magnitude0)

ˆcos22( Sinceˆˆ-=Ð+=

jjeHeH 4 19

PLOT of FREQ RESPONSE

wwwˆˆ)ˆcos22()(jjeeH-+=RESPONSE at pppp/3 }1,2,1{}{=kb (radians)ˆwp-p wˆ 20

EXAMPLE 6.2

y[n]x[n])(ˆwjeH wwwˆˆ)ˆcos22()(jjeeH-+= njjjeenxeHny)3/(4/ˆ2][ andknown is )( when][ Findppw= 21
njjeenxny)3/(4/2][ when][ Findpp=

EXAMPLE 6.2 (answer)

3/ˆat )( evaluate-Step One ˆpww=jeH

wwwˆˆ)ˆcos22()(jjeeH-+=

3/ˆ@3)(3/ˆpwpw==-jjeeH

22

EXAMPLE: COSINE INPUT

)cos(2][ andknown is )( when][ Find43ˆppw+=nnxeHny j wwwˆˆ)ˆcos22()(jjeeH-+= y[n]x[n])(ˆwjeH wˆwˆ 23

EX: COSINE INPUT

)cos(2][ when][ Find43pp+=nnxny ][][][)cos(221)4/3/()4/3/(

43nxnxnxeen

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