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College of Information & Communications

The world goes wireless! Prepared by Sung Ho Cho

Hanyang University

Baseband Demodulation/Detection

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Signals and Noise

Receiver Goal:To retrieve the bit stream from the received waveform as error free as possible.

Two primary causes for error‐performance degradation:The effect of filtering at the transmitter, channel, and receiver

Ex) Inter‐symbol interference (ISI)The effect of electrical noise and interference produced by a variety of sources

Ex) AWGN, Intermodulation noise, Fading, Multiple access interference

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Demodulation & Detection

Demodulate & Sample Detect

WaveformRecovery

(Matched Filter)

TransmittedSignal

DesiredSignal

Component

NoiseComponent

Channel AWGNWith

ChannelCoding

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Detection in AWGN Channel

The pdf of the Gaussian random noise n0:

Conditional pdf’s for detection:

“Likelihood of s1” “Likelihood of s2”

A

-A

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A Vector View of Signals and Noise (1/3)N‐dimensional Orthogonal Space:

N linearly independent functions (or basis functions)

Euclidean distance measurements can be used for detection process.

N=3

“Mutually perpendicular to each other from a geometric point of view!”

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A Vector View of Signals and Noise (2/3)

An arbitrary M‐dimensional finite set of waveforms {si(t)} (i=1, 2, …, M) can be spanned by N‐dimensional basis functions such that

The set of signal waveforms {si(t)} can be viewed as a set of vectors {si}, where

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A Vector View of Signals and Noise (3/3)

Reference signal vectors: sj and skReceived vector: sj+n and sk+nThe receiver knows, a priori, the locations of the reference signal vectors in the signal space.

The task of the receiver:For a received signal vector r,the receiver needs to decide whether r has a close resemble to sj, or whether it more closely resembles sk, or whether it is closer to some other reference vectors in the M‐ary set.

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Orthogonal Representation of Waveform: An Example

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Representation of White Noise with Orthogonal Waveforms

The AWGN can be partitioned into two components:

The noise within the signal space:

The noise vector:

Noise within the signal space, i.e., the projection of the noise components on the signal space

Noise outside the signal space

( )n t

N‐dimensional signal spacespanned by

ˆ( )n t

( )n t

{ }1 2ψ ( ),ψ ( ), ,ψ ( )Nt t t

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Variance of White Noise

Variance (or average noise power) of AWGN:

However, the variance of filtered AWGN becomes finite.For example, the variance of the correlator output is given by

We will assume that the noise of interest in the detection process is the output of a correlator (or matched filter) with variance

Power spectral density of white noise:

(See Appendix C for proof.)

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Eb/N0: The Basic SNR Parameter in Digital Systems

S/N (SNR): average signal power to average noise power ratio

Eb/N0: bit energy to noise power spectral density ratioEnergy‐related metric at the bit levelTb: bit intervalR: bit rateW: bandwidth

PB versus Eb/N0PB: bit error probability

or

“Waterfall‐like shape”

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Why Eb/N0?

For the case of analog communication systems:An analog waveform is classified as a power signal.The energy is not a useful way of characterizing the waveform, since an infinitely long waveform has an infinite amount of energy. The power (i.e., rate of delivering the energy) is a more useful parameter for characterizing analog waveforms.

For the case of digital communication systems:A symbol is transmitted by using a corresponding transmission waveform within a symbol interval.The power is not a useful way of characterizing the waveform, since the power (averaged over all time) goes to zero when focusing onone symbol.The symbol energy (i.e., power integrated over the symbol interval) is a more useful parameter for characterizing digital waveforms.

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Detection of Binary Signals in Gaussian Noise

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Maximum Likelihood Receiver Structure

Likelihood ratio test:

Minimum error criterion:Choose γ=γ0 in a way that the probability of error is minimized.If P(s1) = P(s2) and if the likelihoods, p(z|si) (i=1, 2), are symmetrical,

Maximum likelihood detector:Choose s1(t) if p(za|s1) > p(za|s2).Otherwise, choose s2(t).

The a priori probability that s2(t) is transmitted

The a priori probability that s1(t) is transmitted

Optimum Threshold

(See Appendix B.3.1 for proof.)

p(za|s1)

p(za|s2)

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Bit Error Probability

Conditional bit error probabilities:

The (average) bit error probability:

For equally likely Gaussian noise channel,

or

or

and

where

and

Therefore,

which simplifies to

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Matched Filter (1/5)

The matched filter is the optimum system for detecting a known signal in AWGN.

A linear receiver model:

Problem:Given the received signal r(t), we want to design h(t) in order to detect the pulse signal s(t) in an optimum manner.

Message

ChannelNoise

PSD = N0/2

ai(t) = s(t) * h(t)n0(t) = n(t) * h(t)

s(t)

n(t)

r(t) z(t) z(T)

r(t) = s(t) + n(t)

z(T) = ai + n0

z(t) = ai(t) + n0(t)

H(f)

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Instantaneous signal‐to‐average noise power ratio at t = T:

Signal at the filter output:

Output noise power:

Therefore,

The problem now becomes how to find H(f) = H0(f) for which (S/T)T is maximized.

{ }2 var ( )i ia s t=

{ }20 0var ( )n tσ =

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Schwarz’s inequality:For arbitrary two complex functions f1(x) and f2(x) with finite energy, the following inequality holds:

The equality holds if and only if , where k is any constant.

Applying the Schwarz’s inequality to our case:

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Matched Filter (4/5)Therefore,

Note that the maximum output (S/T)T depends on the input signal energy and the power spectral density of the noise, not on the particular shape of the waveform being used.

Now, the equality holds only if the optimum filter transfer function H0(f) becomes

Since s(t) is real, we obtain

The impulse response of the optimum filter is a time‐reversed and delayed version of the input signal s(t) which is “matched” to the input signal.

where the energy E of the input signal s(t) is

or

or

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Example: Matched filter for a rectangular pulse

Maximum value at t = T

Rectangular pulse

Matched filter output

Integrator output

Integrate-and-dump circuit

Same waveform

Matched filter h(t)

Return to initial condition,

i.e., “dump”

s(t)

si(t)

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Correlation Realization of the Matched Filter

The output of the matched filter:

Substituting to the above equation yields

When t = T, we have

At the receiver, the received signal r(t) is correlated with each reference signal si(t) (i = 1, …, M), using a bank of M correlators.We identify the signal si(t) whose correlation value yields the maximum zi(T) as the transmitted signal.

“Correlation of r(t) and s(t)”

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Optimizing Error Performance (1/5)

Recall that for equally likely Gaussian noise channel, the (average) bit error probability is given by

For minimizing PB, we need to determine the linear filter that maximizes (a1 ‐ a2)/2σ0, or equivalently,

where

Instantaneous power of the difference signal, s1(t)‐s2(t)

Output noise power

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Recall also that the maximally achievable output SNR of the matched filter was given by

Consider the case that the filter is matched to the input difference signal [s1(t)‐s2(t)].

We can then write the output SNR at time t = T as

where N0/2 is the two‐sided PSD of the noise at the filter input, and

By maximizing the output SNR provided above, the matched filter provides the maximum distance between the two outputs, i.e., signal a1 and signal a2.

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Therefore, the optimum bit‐error‐probability which maximizes the output SNR becomes

Matched filter for optimum error performance

Or, equivalent correlator

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We are able to find more general relationship for PB.

Define a cross correlation coefficient:

Note that

Therefore, we have PB in a more general form as

where

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Example 1: Antipodal signals (ρ = −1, θ = 1800)

Example 2: Orthogonal signals (ρ = 0, θ = 900)

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Error Probability Performance of Binary Signaling (1/3)

Unipolar signaling:

Correlator detector for unipolar signaling

( )2 2

22 2

0

1 02 2

d

b

E A T A T

A TE A T T

= − =

⎡ ⎤= ⋅ + ⋅ =⎣ ⎦

22 21 2

01γ 0

2 2 2a a A TA T T+ ⎡ ⎤= = + =⎣ ⎦

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Bipolar signaling:

Correlator detector for bipolar signaling

( )2 2

2 2 2

( ) 41 ( )2

d

b

E A A T A T

E A T A T A T

= − − =

⎡ ⎤= ⋅ + − ⋅ =⎣ ⎦

2 21 20

1γ 02 2

a a A T A T+ ⎡ ⎤= = − =⎣ ⎦

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Bit error performance of unipolar and bipolar signaling:

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Intersymbol Interference (ISI) (1/2)

Consider a binary PAM system (We assume zero delay):

Amplitude of PAM wave:

Transmitted signal:

Received signal:

( ) ( )( ) ( )k b k bk k

s t a t kT g t a g t kTδ⎛ ⎞= − ∗ = −⎜ ⎟⎝ ⎠∑ ∑

1, if symbol is 11, if symbol is 0

kk

k

ba

b+⎧

= ⎨−⎩

The ISI often arises when the communication channel is dispersive.

Transmitter Channel Receiver

Dispersive Matchedfilter

( )

( ) ( ) ( ) ( )

( ) ( ) ( )k bk

x t s t h t w t

a t kT g t h t w tδ

= ∗ +

⎛ ⎞= − ∗ ∗ +⎜ ⎟⎝ ⎠∑

(continued)

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Intersymbol Interference (ISI) (2/2)

Receiver filter output:

Sampled version of y(t) at time ti = iTb:

In the absence of both ISI and noise, we have

We need to minimize the effects of noise and ISI.

( )

( ) { }

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

k bk

k bk

k bk

y t x t c t

a t kT g t h t w t c t

a t kT g t h t c t n t

a p t kT n t

δ

δ

μ

= ∗

⎧ ⎫⎛ ⎞= − ∗ ∗ + ∗⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

⎛ ⎞= − ∗ ∗ ∗ +⎜ ⎟⎝ ⎠

= − +

∑

∑

∑

( )

( )

( ) ( )

( )

i k b ik

i k b ikk i

y t a p i k T n t

a a p i k T n t

μ

μ μ∞

=−∞≠

= − +⎡ ⎤⎣ ⎦

= + − +⎡ ⎤⎣ ⎦

∑

∑

Intersymbol interference (ISI)Contribution of the i-th transmitted bit

( )i iy t aμ=

( ) ( ) ( ) ( )p t g t h t c tμ = ∗ ∗

Dispersive channel ISI

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Eye Patterns (1/4)

The eye pattern is defined as the synchronized superposition of all possible realizations of the signal of interest viewed within a particular signaling interval.

Eye opening,i.e., region without ISI

Location of the symbol

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Eye Patterns (2/4)

Example: Eye pattern for a noiseless quaternary system:

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Eye Patterns (3/4)

Eye pattern for a quaternary system with SNR = 20 dB:

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Eye Patterns (4/4)

Eye pattern for a quaternary system with SNR = 10 dB:

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