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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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