Wireless PHY: Frequency-Domain Analysis

Y. Richard Yang

09/4/2012

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Outline

Admin and recap Frequency domain analysis (Fourier

series)

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Admin

Slides and reading posted to class home page

It is important to read the assigned reading for today’s class

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Recap: Wireless and Mobile Computing

Driven by infrastructure and device technology global infrastructures device miniaturization and capabilities software development platforms

Challenges: wireless channel: unreliable, open access mobility portability changing environment heterogeneity

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Overview of Wireless Physical Layer

source decoding

bitstream

channel decoding

receiver

demodulation

source coding

bitstream

channel coding

analogsignal

sender

modulation

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Wireless Physical Layer Example: Wireless: 802.11

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Understand key issues and techniques in the design of wireless physical layer

Key approach: identify the problem and then the solution(s).

Our Objective

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Outline

Recap Frequency domain analysis (Fourier

series)

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1

0

t

periodical signal

Fourier Series: Decomposing into a Collection of Harmonics

Time domain 1

0

t

decomposition

A periodic real function g(t) on [-π, π] can be decomposed as a set of harmonics (cos, sin):

set bk = 0

Fourier Series

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Fourier Series

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Fourier Series: Example

http://en.wikipedia.org/wiki/File:Periodic_identity_function.gif

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Fourier Series: An Alternative Representation

A problem of the expression

It contains both cos() and sin() functions, and hence is somehow complex to manipulate.

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Fourier Series: Using Euler’s formula

Applying Euler’s formula

We have

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Fourier Series: Using Euler’s formula

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Making Sense of Complex Numbers

What is the effect of multiplying c by ejπ/2?

What is the effect of multiplying c by j?

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Making Sense of Complex Numbers

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Making Sense of Complex Numbers: Conjugate

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Summary of Progress: Fourier Series of Real Function on [-π, π]

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Defining Decomposition on a General Interval

A periodic function g(t) with period T on [a, a+T] can be decomposed as:

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Defining Decomposition on [0, 1]

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Making Sense of ej2πftej2πft

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Making Sense of ej2πft

ej2πft

ϕ=2πft

e-j2πft

ϕ=-2πft

G[-f]e-j2πft

G[f]ej2πft

Two Domain Representations

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- Two representations: time domain; frequency domain- Knowing one can recover the other

Example: Frequency Seriesof sine and cosine

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Example: Frequency Seriesof sine and cosine

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Example: Frequency Domain’s View of Euler’s Formula

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Example: Frequency Domain’s View of Euler’s Formula

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Example: Frequency Domain’s View of Euler’s Formula

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From Integral to Computation

Discrete Domain Analysis

Transforming a sequence of numbers x0, x1, …, xN-1 to another sequence of numbers X0, X1, …, XN-1

Inverse DFFT

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Two Notes on Sampling

How fast to sample a time signal? According to Nyquist’s Law, if the highest

frequency is W, you need to sample at least 2W samples/sec

How to map from FFT output frequency when applied to samples? Assume discrete FFT applies to Nfft samples

The interpretation is that Nfft samples is 1 sec

Hence, FFT output of X1 is for the base frequency

But Nfft is only Nfft/Nsample sec => X1 is for frequency of Nfft/Nsample

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Frequency Domain Examples Using GNURadio

spectrum_samples Observe sample/under sample

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Frequency Domain Examples Using GNURadio

spectrum_2sin_plus Audio FFT Sink Scope Sink Noise

Backup Slides

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Implementing Wireless: From Hardware to Software