lFILTERING Lec5

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M.H. Perrott 2007 Filtering in Continuous and Discrete Time, Slide 1

Filtering inContinuous andDiscrete Time

Lowpass, highpass, bandpass filtering Filter response to cosine wave inputs Discrete-Time Fourier Transform Filtering based on difference equations

Copyright 2007 by M.H. PerrottAll rights reserved.


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0

X(f)

Voiceband

frequencies

f

RF transmission

frequencies

0

Y(f)

Voiceband

frequencies

f

RF reception

frequencies

RF

Transmitter

Voice

RF

Transmission

RF

Receiver

Voice

RF

Reception

Interferer

Motivation for Filtering

Filtering is used to remove undesired signalsoutside of the frequency band of interest Enables selection of a specific radio, TV, WLAN, cell

phone, cable TV channel Undesired channels are often called interferers


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1

0 ffo

R(f)

-fo 2fo-2fo

2A

0f

fo

W(f)

-fo

A A

2fo-2fo

2A

0f

fo

H(f)

-fo 2fo-2fo

Lowpass Filter

Only low frequency(i.e. baseband)portion remains

y(t) = 2cos(2fot)

z(t) r(t)

LowpassFilter

w(t)H(f)


4/21M.H. Perrott 2007 Filtering in Continuous and Discrete Time, Slide 4

1

0 ffo

R(f)

-fo 2fo-2fo

0f

fo

W(f)

-fo

A A

2fo-2fo

2A

0f

fo

H(f)

-fo 2fo-2fo

A A

1

Highpass Filter

Only high frequency(i.e. RF) portionremains

y(t) = 2cos(2fot)

z(t) r(t)

HighpassFilter

w(t)H(f)



1

0 ffo

R(f)

-fo 2fo-2fo

0f

fo

W(f)

-fo

A A

2fo-2fo

2A

0f

fo

H(f)

-fo 2fo-2fo

A A

1

Bandpass Filter

Only high frequency(i.e. RF) portionremains

y(t) = 2cos(2fot)

z(t) r(t)

BandpassFilter

w(t)H(f)



1

0 ffo

R(f)

-fo 2fo-2fo

0f

fo

W(f)

-fo

A A

2fo-2fo

2A

0f

fo

H(f)

-fo 2fo-2fo

A A

1

Interferer

Only desiredRFportion remains Interferer eliminated

Why is Bandpass Filtering Useful? Allows removal of interfering signals

Highpass filtering would be of limited use here

Typically higher complexity implementation than withlowpass or highpass filters Many RF systems such as cell phones use specialized

components called SAW filtersto achieve bandpass filtering



ffc

K

-fc

K2

1

0ffc

K

-fc 0

Ideal Lowpass Practical Lowpass

H(f)

H(f)

ffc-fc

H(f)

fc

-fc

H(f)

A More Formal Treatment of Filters

An ideal filter would have a brickwall magnitude

response and zero phase response Practical filters have a more gradual magnitude rolloffand a non-zero phase response

Design of the filter usually focuses on getting a

reasonable magnitude rolloff with a specifiedcutoff frequency fc (i.e., filter bandwidth)



ffo

K

-fo 0

H(f)

f

H(f)

0

1

ffo

X(f)

1

-fo

fo-fo

Lowpass Filter: H(f)

0 ffo

Y(f)

-fo

H(fo)H(fo)

ej

H(-fo)H(-fo)

ej

Response of Filter to Input Cosine

Fourier transform analysis:

Key properties of practical filters Magnitude response is even:

Phase response is odd:

Well explain why this is true in 6.003


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Compute Fourier Transform

Fourier transform of output:

Define:

ffo

K

-fo 0

H(f)

f

H(f)

0

1

ffo

X(f)

1

-fo

fo-fo


0 ffo

Y(f)

-fo

H(fo)H(fo)

ej

H(-fo)H(-fo)

ej


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Compute Time-Domain Response

Transform back to time domain:

ffo

K

-fo 0

H(f)

f

H(f)

0

1

ffo

X(f)

1

-fo

fo-fo


0 ffo

Y(f)

-fo

H(fo)H(fo)

ej

H(-fo)H(-fo)

ej


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ffo

K

-fo 0

H(f)

f

H(f)

0

1

ffo

X(f)

1

-fo

fo-fo


0 ffo

Y(f)

-fo

H(fo)H(fo)

ej

H(-fo)H(-fo)

ej

Key Observations of Filter Response

Input cosine wave is scaledin amplitude and

phase-shiftedin time Scale factor set by magnitude of H(f)at f=fo Phase shift set by phase of H(f)at f=fo

We typically focus only on the magnitudeof thefrequency response, H(f), of the filter


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Designing and Using Filters Within Matlab

Our lab exercises will have you design and use

filters in Matlab Matlab will interface to the USRP board in order toreceive real world signals from the antenna

Matlab framework is based on discrete-timesequences(which are indexed on integer values) Correspond to samplesof corresponding real world signals

(which are continuous-timein nature)

n

x[n]

1

x(t)

t

Real World Matlab

We need another Fourier analysis tool


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The Discrete-Time Fourier Transform

Allows us to deal with non-periodic, discrete-timesignals

Frequency domain signal isperiodicin this case

Where:

n

x[n]

1

0

X(ej2)

1-1

Note: fftfunction in Matlab used to compute DTFT


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n

x[n]

1

0

X(ej2)

1-1

t

x(t)

T

0f

X(f)

1/T-1/T

Matlab Sequence Samples of Real World Signal

Relating to Samples of `Real World Signals

Samples of a continuous-time signal with sampleperiod Tleads to frequency domain signal withperiod 1/T

We simply scale frequency axis of fftin Matlab We will say much more about samplinglater


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Filters Within Matlab

Implemented as difference equations Current output, y[n], depends on weighted values of

previous output samples and current and previous input

samples, x[n]

Group aand bcoefficients as vectors:

Execute filter using the filtercommand:

Discrete-Time

Filterx[n]H(ej2)

n

x[n]

1

y[n]

n

1

y[n]


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Impact of Delay on DTFT

Consider a signal that is a delayed version ofanother signal:

Compute DTFT of y[n]


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Compute Filter Response using DTFT

Make use of the time shift property:

Filter response is simply ratio of output over input:

F F l


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FIR Filters

Finite Impulse Response (FIR) filters use only bcoefficients in their implementation

Example:

x[n]H(ej2)

y[n]

k

bk

Nk

ak

M1

1

0

Note:

Fil O d f FIR Fil


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Filter Order for FIR Filters

Consider two different values for N

Higher Nleads to steeper filter response We refer to Nas the orderof the filter

H(ej2)

0 1-1

8

H(ej2)

0 1-1

4

N=3 N=7

x[n]

H(e

j2

)

y[n]

k

bk

Nk

ak

M1

1

0

FIR Filt D i i M tl b


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FIR Filter Design in Matlab

Lowpass, highpass, and bandpass filters can berealized by appropriately scaling the relative valueof the bcoefficients Higher order (i.e., higher N) leads to steeper responses

Perform FIR filter design using fir1command

Frequency response observed with freqzcommand

x[n]

H(ej2

)

y[n]

k

bk

Nk

ak

M1

K

0

See Prelab portion of Lab 3 for details

S


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Summary Filters can generally be classified according to

Lowpass, highpass, bandpass operation

Bandwidth and order of filter

Given a cosine input to a filter, output is: Scaled in amplitude by magnitude of filter frequency response

Shifted in phase by phase of filter frequency response

Matlab operates on discrete-time signals Use DTFT for analytical analysis

Use commands such as fir1, freqz, and filterfor design andimplementation of FIR filters

Next lecture: introduce I/Q modulation and furtherdiscuss continuous-time filtering

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