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Information Transmission

Chapter 2, repetition FREDRIK TUFVESSON

ELECTRICAL AND INFORMATION TECHNOLOGY

Linear, time-invariant (LTI) systems

A system is said to be linear if,

whenever an input yields an output

and an input yields an output

We also have

where are arbitrary real or complex constants.

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What does this mean?

What does this mean

• Input zero results in output zero for all linear systems!

Superposition:

• the output resulting from an input that is a weighted sum

of signals

is the same as

• the weighted sum of the outputs obtained when the input

signals are acting separately.

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The delta function

When the duration of our pulse approaches 0, the pulse

approaches the delta function

(also called Dirac's delta function or, the unit impulse)

Properties of the delta function

The delta function is defined by the property

where g(t) is an arbitrary function, continuous at the origin.

The derivative of the step function is the delta

The integral of the delta is the step function

t

g(0)

g(t)

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The output of LTI systems

The integral is called convolution and is denoted

The output y(t) of a linear, time-invariant system is the

convolutional of its input x(t) and impulse response h(t).

Example 3

Consider an LTI system with impulse response

and input

What is the output?

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The transfer function

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The transfer function

is called the frequency function or the transfer function for

the LTI system with impulse response h(t).

Phase and amplitude functions

The frequency function is in general a complex function of

the frequency:

where

is called the amplitude function and

is called the phase function.

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The Fourier transform

The Fourier transform

The Fourier transform of the signal x(t) is given by the

formula

This function is in general complex:

where is called the spectrum of x(t)

and its phase angle.

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Spectrum of a consine

Hence we have a Fourier transform pair

Properties of the Fourier transform

1. Linearity

2. Inverse

3. Translation (time shifting)

4. Modulation (frequency shifting)

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Properties of the Fourier transform

5. Time scaling

6. Differentiation in the time domain

7. Integration in the time domain

8. Duality

Properties of the Fourier transform

9. Conjugate functions

10. Convolution in the time domain

11. Multiplication in the time domain

12. Parseval's formulas

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Fourier transform of a convolution

Since the output y(t) of an LTI system is the convolution of

its input x(t) and impulse response h(t) it follows from

Property 10 (Convolution in the time domain) that the

Fourier transform of its output Y(f) is simply the product of

the Fourier transform of its input X(f) and its frequency

function H(f), that is,

Some useful Fourier transform pairs

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Some useful Fourier transform pairs

Sampling and signals

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Orthogonal functions

Orthogonality is an important notion in signal analysis. it

means that

where is the energy of

Normalized functions

Furthermore, since

for all k, these functions are normalized (energy ).

A set of orthogonal and normalized functions is called an

orthonormal set of functions.

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The sampling theorem

If x(t) is a signal whose Fourier transform is

identically zero for , then x(t) is completely

determined by its samples taken every

seconds in the manner

Nyquist rate/Nyquist frequency

The sample points are taken at the rate 2W

samples per second.

If W is the smallest frequency such that the Fourier

transform of x(t) is identically zero for then the

sampling rate 2W is called the Nyquist rate or Nyquist

frequency.

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Channels

Free-space loss, Friis’ law

Received power, with antenna gains GTX and GRX:

2

4

RX TXRX TX TX RX TX

free

G GP d P P G G

L d d

| | | | |

2

| | 10 |

410log

RX dB TX dB TX dB free dB RX dB

TX dB TX dB RX dB

P d P G L d G

dP G G

In free space, the received power

decays with the distance squared

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Receiver noise: Noise sources

The power spectral density of a noise source is usually given in one

of the following three ways:

1) Directly [W/Hz]:

2) Noise temperature [Kelvin]:

The power of the noise is also determined by the bandwidth

where k is Boltzmann’s constant (1.38x10-23 W/Hz) and Tk is the

room temperature of 290 K (17o C).

Analog modulation

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Analog modulation

• Shift the frequency to an appropriate frequency for

transmission

• Vary the amplitude or phase to represent the information

– Constant phase variation=frequency shift

• The original signal is often called the baseband signal

Modulation property

• Shifting the frequency does not modify the information

content

• There are two replicas, one at positive frequencies and

one at negative

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Frequency modulation

• Idea: Let the baseband signal change the frequency of

the bandpass signal

– High amplitude – high frequency

– Low amplitude – low frequency

• The frequency of a signal is the derivative of its phase

Digital modulation

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Some basics

• Each bit, or groups of bits, is represented by an analog

waveform v(t)

• Symbol rate Rt=1/T

• Symbol energy Es

• The power of the signal is given by Es/T = Es R

• Data an,

4ASK

4PSK

4FSK

A t

00 01 11 00 10

00 01 11 00 10

00 01 11 00 10

- Amplitude carries information

- Phase constant (arbitrary)

- Amplitude constant (arbitrary)

- Phase carries information

- Amplitude constant

(arbitrary)

- Phase slope (frequency)

carries information

Comment:

Amplitude, phase and frequency

modulation

𝑠 𝑡 = 𝐴 𝑡 𝑐𝑜𝑠 2𝜋𝑓𝑐𝑡 + 𝜑(𝑡)

𝝋(𝒕)

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Complex domain

The modulation process

Mapping PAM mb mc LPs t

exp 2 cj ft

Re{ }

Radio

signal

PAM:

“Standard” basis pulse criteria

(energy norm.)

(orthogonality)

Complex

numbers

Bits

Symbol

time

=m

sm mTtvc=tsLP

12

=dttv

0for 0*

m=dtmTtvtv s

Basis pulses

(Root-) Raised-cosine [in freq.]

Rectangular [in time]

TIME DOMAIN FREQ. DOMAIN

sTf freq. Normalized

Normalized freq. f × T s

sT t/ timeNormalized

sT t/ timeNormalized

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A golden bandwidth rule

The narrowest bandwidth of any

pulses that act independently is

[-1/2T, 1/2T]

where T is the symbol interval

Optimal receiver

To be able to better measure the “fit” we look at the energy of the

residual (difference) between received and the possible noise free signals:

t

r(t), s0(t)

0:

t

r(t), s1(t)

1: t

s1(t) - r(t)

t

s0(t)-r(t)

2

1s t r t dt

2

0s t r t dt

This residual energy is much

smaller. We assume that “0”

was transmitted.

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Optimal receiver

Bit-error rates (BER)

2PAM 4QAM 8PSK 16QAM

Bits/symbol 1

Symbol energy Eb

BER 0

2 bEQ

N

2

2Eb

0

2 bEQ

N

3

3Eb

0

20.87

3

bEQ

N

4

4Eb

,max

0

3

2 2.25

bEQ

N

EXAMPLES:

0 2 4 6 8 10 12 14 16 18 20 10

-6

10 -5

10 -4

10 -3

10 -2

10 -1

10 0

Optimal receiver

Bit-error rates (BER), cont.

0/ [dB]bE N

Bit

-err

or

rate

(B

ER

)

2PAM/4QAM

8PSK

16QAM

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