Wireless PHY: Modulation and Demodulation

Y. Richard Yang

09/11/2012

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Outline

Admin and recap Amplitude demodulation Digital modulation

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Admin

Assignment 1 posted

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Objectiveo Frequency assignment

Basic conceptso the information source (also called baseband)o carriero modulated signal

Recap: Modulation

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Recap: Amplitude Modulation (AM)

Block diagram

Time domain

Frequency domain

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Recap: Demod of AM

Design option 1: multiply modulated signal by e-jfct, and then LPF

Design option 2: quadrature sampling

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Example: Scanner

Setting: a scanner scans 128KHz blocks of AM radio and saves each block to a file.

For the example file During scan, fc = 710K LPF = 128K (one each side)

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Exercise: Scanner

Requirements Scan the block in a saved file to find radio stations and

tune to each station (each AM station has 10 KHz) Audio device requires 48K sample rate for playback

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Remaining Hole: How to Design LPF

Frequency domain view

freqB-B

freqB-B

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Design Option 1

freqB-B

freqB-B

compute freq

zeroing outoutband freq

compute lower-passtime signal

This is essentially how image compression works.

Problem(s) of Design Option 1?

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Design Option 2: Impulse Response Filters

GNU software radio implements filtering using Finite Impulse Response (FIR) filters Infinite Impulse Response (IIR) Filters FIR filters are more commonly used

FIR/IIR is essentially online, streaming algorithms

They are used in networks/communications/vision/robotics…

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FIR Filter

An N-th order FIR filter h is defined by an array of N+1 numbers:

They are often stored backward (flipped)

Assume input data stream is x0, x1, …,

h0h1h2hN…

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FIR Filter

xnxn-1xn-2xn-3

h0h1h2h3

****

xn+1

compute y[n]:

3rd-OrderFilter

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FIR Filter

xnxn-1xn-2xn-3

h0h1h2h3

****

xn+1

compute y[n+1]

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FIR Filter

is also called convolution between x (as a vector) and h (as a vector), denoted as

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Key Question Using h to Implement LPF

Q: How to determine h?

Approach: Understand the effects of y=g*h in the

frequency domain

g*h in the Continuous Time Domain

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Remember that we consider x as samples of time domain function g(t) on [0, 1] and (repeat in other intervals)

We also consider h as samples of time domain function h(t) on [0, 1] (and repeat in other intervals)

for (i = 0; i< N; i++) y[t] += h[i] * g[t-i];

Visualizing g*h

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g(t)

h(t)

time

0 T 0T

Visualizing g*h

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g(t)

h(0)

timet

0 T 0T

g(t)

Fourier Series of y=g*h

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Fubini’s Theorem

In English, you can integrate first along y and then along x first along x and then along y at (x, y) gridThey give the same result

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See http://en.wikipedia.org/wiki/Fubini's_theorem

Fourier Series of y=g*h

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Summary of Progress So Far

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y = g * h => Y[k] = G[k] H[k]

In the case of Fourier Transform, y = g * h => Y[f] = G[f] H[f]

is called the Convolution Theorem, an important theorem.

Applying Convolution Theorem to Design LPF

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Choose h() so that H() is close to a rectangle shape

h() has a low order (why?)

f1/2-1/2

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Sinc Function

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The h() is often related with the sinc(t)=sin(t)/t function

f1/2-1/2

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FIR Design in Practice

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Compute h MATLAB or other design software GNU Software radio: optfir (optimal filter

design) GNU Software radio: firdes (using a method

called windowing method)

Implement filter with given h freq_xlating_fir_filter_ccf or fir_filter_ccf

LPF Design Example

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Design a LPF to pass signal at 1 KHz and block at 2 KHz

LPF Design Example

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#create the channel filter # coefficients chan_taps = optfir.low_pass( 1.0, #Filter gain 48000, #Sample Rate 1500, #one sided mod BW (passband edge) 1800, #one sided channel BW (stopband edge) 0.1, #Passband ripple 60) #Stopband Attenuation in dB print "Channel filter taps:", len(chan_taps) #creates the channel filter with the coef foundchan = gr.freq_xlating_fir_filter_ccf( 1 , # Decimation rate chan_taps, #coefficients 0, #Offset frequency - could be used to shift 48e3) #incoming sample rate

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Outline

Recap Amplitude demodulation

frequency shifting low pass filter

Digital modulation

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Modulation of digital signals also known as Shift Keying

Amplitude Shift Keying (ASK): vary carrier amp. according to data

Frequency Shift Keying (FSK)o vary carrier freq. according to bit value

Phase Shift Keying (PSK)o vary carrier freq. according to data

1 0 1

t

1 0 1

t

1 0 1

t

Modulation

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BPSK (Binary Phase Shift Keying): bit value 1: cosine wave cos(2πfct)

bit value 0: inverted cosine wave cos(2πfct+π)

very simple PSK Properties

robust, used e.g. in satellite systems

Q

I

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Phase Shift Keying: BPSK

one bit time T

1

one bit time T

0

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Phase Shift Keying: QPSK

Q

I

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01

10

00

QPSK (Quadrature Phase Shift Keying): 2 bits coded at a time we call the two bits as one symbol symbol determines shift of cosine

wave often also transmission of relative,

not absolute phase shift: DQPSK - Differential QPSK

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Quadrature Amplitude Modulation (QAM): combines amplitude and phase modulation

It is possible to code n bits using one symbol 2n discrete levels

0000

0001

0011

1000

Q

I

0010

φ

a

Quadrature Amplitude Modulation

Example: 16-QAM (4 bits = 1 symbol)

Symbols 0011 and 0001 have the same phase φ, but different amplitude a. 0000 and 1000 have same amplitude but different phase

Generic Representation of Digital Keying (Modulation) Sender sends symbols one-by-one M signaling functions g1(t), g2(t), …, gM(t),

each has a duration of symbol time T Each value of a symbol has a signaling

function

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Exercise: gi() for BPSK

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1: g1(t) = cos(2πfct) t in [0, T]

0: g0(t) = -cos(2πfct) t in [0, T]

Are the two signaling functions independent? Hint: think of the samples forming a vector, if

it helps, in linear algebra Ans: No. g1(t) = -g0(t)

cos(2πfct)[0, T]1-1

Q

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g1(t)g0(t)

Exercise: Signaling Functions gi() for QPSK

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11: cos(2πfct + π/4) t in [0, T]

10: cos(2πfct + 3π/4) t in [0, T]

00: cos(2πfct - 3π/4) t in [0, T]

01: cos(2πfct - π/4) t in [0, T]

Are the four signaling functions independent? Ans: No. They are all linear combinations of sin(2πfct) and

cos(2πfct).

Q

I

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01

10

00

QPSK Signaling Functions as Sum of cos(2πfct), sin(2πfct)

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11: cos(π/4 + 2πfct) t in [0, T]-> cos(π/4) cos(2πfct) +

-sin(π/4) sin(2πfct)

10: cos(3π/4 + 2πfct) t in [0, T]-> cos(3π/4) cos(2πfct) +

-sin(3π/4) sin(2πfct)

00: cos(- 3π/4 + 2πfct) t in [0, T]-> cos(3π/4) cos(2πfct) +

sin(3π/4) sin(2πfct)

01: cos(- π/4 + 2πfct) t in [0, T]-> cos(π/4) cos(2πfct) +

sin(π/4) sin(2πfct)

sin(2πfct)

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00

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cos(2πfct)

[cos(π/4), sin(π/4)]

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[cos(3π/4), sin(3π/4)]

[cos(3π/4), -sin(3π/4)]

[-sin(π/4), cos(π/4)]

We call sin(2πfct) and cos(2πfct) the bases.

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Outline

Recap Amplitude demodulation

frequency shifting low pass filter

Digital modulation modulation demodulation

Key Question: How does the Receiver Detect Which gi() is Sent?

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Assume synchronized (i.e., the receiver knows the symbol boundary).

Starting Point

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Considered a simple setting: sender uses a single signaling function g(), and can have two actions send g() or nothing (send 0)

How does receiver use the received sequence x(t) in [0, T] to detect if sends g() or nothing?

Design Option 1

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Sample at a few time points (features) to check

Issue Not use all data points, and less robust to

noise

Design Option 2

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Streaming algorithm, using all data points in [0, T] As each sample xi comes in, multiply it by a factor hT-i-

1 and accumulate to a sum y

At time T, makes a decision based on the accumulated sum at time T: y[T]

xTx2x1x0

h0h1h2hT

****

Example Streaming (Convolution/Correlation):

Assume incoming x is a rectangular pulse (in baseband) and h is also a rectangular pulse

A gif animation (play in ppt) presentation): redline g(): the sliding filter h(t) blue line f(): the input x()

43Source: http://en.wikipedia.org/wiki/File:Convolution_of_box_signal_with_itself2.gif

Determining the Best h

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where w is noise,

Design objective: maximize peak pulse signal-to-noise ratio

Determining the Best h

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Assume Gaussian noise, one can derive

Using Fourier Transform and Convolution Theorem:

Determining the Best h

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Apply Schwartz inequality

By considering

Determining the Best h

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Determining Best h to Use

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xTx2x1x0

gTg2g1g0

****

xTx2x1x0

h0h1h2hT

****

Matched Filter Decision

is called Matched filter.Example

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decision time

Backup Slides

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Modulation