recimplmatchfil.pdf

Chapter 4

Receiver Implementation, Matched Filters

Summary: We discuss receiver structures for optimum waveform communication here.The optimum receiver can be implemented as a matched-filter receiver, hence a matchedfilter not only maximizes the signal-to-noise ratio but it can also be used to minimize PE .

4.1 Dwight O. North

Figure 4.1: Dwight O. North, inventor of the matched filter. Photo IEEE-IT Soc. Newsl., Dec.1998.

Dwight O. North was one of the recipients of the IEEE Information Society’s Golden JubileeAwards for Technological Innovation. He was cited for his invention of the matched filter. Sadly,Dr. North died on June 26, 1998, just two months before the awards were presented at ISIT’98at MIT.

The importance of the matched filter concept in communications and signal processing hardlyneeds to be repeated. Dr. North was the first to formalize this concept, which he published in

59

60 CHAPTER 4. RECEIVER IMPLEMENTATION, MATCHED FILTERS

a 1943 classified report [14] at RCA Labs in Princeton. (North did not use the name “matchedfilter”. This term was coined by David Middleton and J.H. Van Vleck, who independently pub-lished the result a year after North in a classified Harvard Radio Research Lab report [24].)North’s report was later reprinted in the Proceedings of the IEEE, in July 1963. This remark-able report introduced not only the matched filter, but also the Rice distribution, the concept offalse alarms to set a detection threshold, studies of pre-detection and post-detection integration,among other topics. Its anticipation of so many of the issues that occupied the attention of radarengineers for many years is quite remarkable.

Dwight North, or Don (for his initials - D.O.N.) as he was known to his friends and col-leagues, was born in Hartford, Connecticut, and was educated at Wesleyan University and atCaltech, from which he received a Ph.D. in Physics in 1933. From 1934 until his retirement in1974, he worked for RCA, first in Harrison, New Jersey, and then as an original member of thetechnical staff at RCA’s Princeton labs when they were established in 1942. His interest in noiseproblems began during the 1930’s when he worked on the study of noise in vacuum tubes oper-ating in the 100MHz band, work being conducted at RCA during its development of commercialtelevision. (His interest in noise problems even extended to the naming of the street on whichhe was a longtime resident in Princeton: Random Road - so named because many of its originalresidents were RCA ”noise” experts, including Dwight North.) During World War II, he workedat the MIT Radiation Lab on the development of radar. After the war, he turned to the studyof solid state physics, which occupied most of the remainder of his career at RCA. (From IEEEIT-Society.)

4.2 Problem DescriptionWe have seen in the previous chapter that an optimal receiver must perform the operations∫

r(t)φi (t)dt for i = 1, · · · , N , to retrieve the received vector r from the received waveformr(t). The problem that we investigate here is whether these operation can be simplified. It turnsout that instead of multiplying and integrating we can also do a filter-operation with a suitablychosen filter.

4.3 IntroductionThe optimum receiver for communication of messages over a waveform channel first determinesr = (r1, r2, · · · , rN ), i.e. the relevant received vector (that was named r1 in section 3.9 of theprevious chapter). For this vector r we have that

ri =∫ ∞

−∞r(t)φi (t)dt for i = 1, 2, · · · , N . (4.1)

This vector r is then used to find the m ∈ M that minimizes (see 2.36)

∥r − sm∥2 − 2σ 2 ln Pr{M = m} = ∥r − sm∥2 − N0 ln Pr{M = m}, (4.2)

4.4. CORRELATION RECEIVER 61

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largest

m̂c2

c1

rN

r2

r1

r(t)×

× integrator

integrator

integrator

×

(r · s1)

(r · s2)

��

��

(r · s|M|)

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c|M|

∑i ri smi

+

+

+

φN (t)

φ2(t)

φ1(t)

selectweightingmatrix

Figure 4.2: The structure of the correlation receiver.

where we substituted N0/2 for σ 2. With

(a · b)1=

∑i=1,N

ai bi , (4.3)

we can rewrite this as follows:

∥r − sm∥2 = (r − sm) · (r − sm)

= (r · r) − 2(r · sm) + (sm · sm) = ∥r∥2 − 2(r · sm) + ∥sm∥2. (4.4)

RESULT 4.1 Since ∥r∥2 does not depend on m, the optimum receiver has to maximize

(r · sm) + cm over m ∈ M = {1, 2, · · · , |M|} (4.5)

with

cm1= N0

2ln Pr{M = m} − ∥sm∥2

2. (4.6)

4.4 Correlation Receiver

The method described in the introduction immediately suggests the receiver structure shown infigure 4.2. There we first see a bank of N multipliers and integrators. This bank yields thevector r . It correlates r(t) with all building-block waveforms. Note that this part of the optimumreceiver also was shown in figure 3.11 and was subject of investigation of the previous chapter.It also appeared as the detector (demodulator) in figure 3.13 at the end of the previous chapter.


After having determined r we use matrix multiplication to obtain the dot products (r · sm) form ∈ M.

(r · s1)

(r · s2)...

(r · s|M|)

=

s11 s12 . . . s1Ns21 s22 . . . s2N...

.... . .

...

s|M|1 s|M|2 . . . s|M|N

r1r2...

rN

. (4.7)

Adding the constants cm and picking the m that achieves the largest sum yields an optimumreceiver.

4.5 Matched-Filter ReceiverIf for all i = 1, N the building-block waveforms are such that φi (t) ≡ 0 for t < 0 and t > T ,then we can replace the N multipliers and integrators by N matched filters and samplers. Thiscan be advantageous in analog implementations since accurate multipliers are then hard to buildwhile filters are more easily designed.

- -ui (t)

hi (t) = φi (T − t)r(t)

Figure 4.3: A filter matched to the building-block waveform φi (t).

Consider (see figures 4.3 and 4.4) a filter with an impulse response hi (t) = φi (T − t) forsome i ∈ M. Note that hi (t) ≡ 0 for t < 0 (i.e. the filter is causal) and also for t > T .

For the output of this filter we then get

ui (t) =∫ ∞

−∞r(α)hi (t − α)dα

=∫ ∞

−∞r(α)φi (T − t + α)dα. (4.8)

t��A

AA

AAA �

��

��A

AAAAA

T T0 0

φi (t) φi (T − t)

t

Figure 4.4: A building-block waveform φi (t) and the impulse response φi (T − t) of the corre-sponding matched filter.

4.6. PARSEVAL RELATIONSHIPS 63

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c|M|

r(t)

r1

r2

rN

c1

c2m̂

largestmatrixweighting select

φ1(T − t)

φ2(T − t)

φN (T − t)

u1(t)

u2(t)

uN (t)

sample at t = T

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∑i ri smi

(r · s1)

(r · s2)

��

��

(r · s|M|)

��

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Figure 4.5: Matched-filter receiver

For t = T the matched-filter output

ui (T ) =∫ ∞

−∞r(α)φi (α)dα = ri , (4.9)

hence we can determine the i-th component of the relevant vector r in this way.A filter whose impulse response is a delayed time-reversed version of a signal φi (t) is called

matched to φi (t). A receiver that is equipped with such filters is called a matched-filter receiver(see figure 4.5).

4.6 Parseval RelationshipsConsider an orthonormal base {φi (t), i = 1, 2, · · · , N } and two waveforms f (t) and g(t) thatcan be expressed in terms of the building-block waveforms that make up this base, i.e.

f (t) 1=∑

i=1,N

fiφi (t),

g(t) 1=∑

i=1,N

giφi (t). (4.10)

The vector-representations that correspond to f (t) and g(t) are

f = ( f1, f2, · · · , fN ) andg = (g1, g2, · · · , gN ). (4.11)


RESULT 4.2 Then∫ ∞

−∞f (t)g(t)dt =

∫ ∞

−∞

∑i=1,N

∑j=1,N

fi g jφi (t)φ j (t)dt

=∑

i=1,N

∑j=1,N

fi g j

∫ ∞

−∞φi (t)φ j (t)dt

=∑

i=1,N

∑j=1,N

fi g jδi j =∑

i=1,N

fi gi = ( f · g). (4.12)

This result says that the correlation of f (t) and g(t), which is defined as the integral of theirproduct, is equal to the dot product of the corresponding vectors. Note that this result can beregarded as an analogue to the Parseval relation in Fourier analysis which says that∫ ∞

−∞f (t)g(t)dt =

∫ ∞

−∞F( f )G∗( f )d f, (4.13)

with

F( f ) =∫ ∞

−∞f (t) exp(− j2π f t)dt and

f (t) =∫ ∞

−∞F( f ) exp( j2π f t)d f. (4.14)

Here G∗( f ) is the complex conjugate of G( f ). The consequences of result 4.2 are:

• Take g(t) ≡ f (t) then ∫ ∞

−∞f 2(t)dt = ( f · f ) = ∥ f ∥2. (4.15)

This means that the energy of waveform f (t) is simply the square of the length of thecorresponding vector f . We therefore also call the squared length of a vector its energy.

Note that now we can rewrite (4.6) as

cm1= N0

2ln Pr{M = m} − ∥sm∥2

2= N0

2ln Pr{M = m} − Em

2, (4.16)

with

Em1=

∫ ∞

−∞s2

m(t)dt, (4.17)

the energy corresponding to the waveform sm(t), for m ∈ M = {1, 2, · · · , |M|}.• Consider∫ ∞

−∞r(t)sm(t)dt =

∫ ∞

−∞r(t)

∑i=1,N

smiφi (t)dt

=∑

i=1,N

smi

∫ ∞

−∞r(t)φi (t)dt =

∑i=1,N

smiri = (sm · r). (4.18)

4.7. SIGNAL-TO-NOISE RATIO 65

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m̂s1(T − t)

s2(T − t)

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s|M|(T − t)

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r(t)

Figure 4.6: Direct receiver.

This is a result similar to (4.12) but not identical since r(t) ̸= ∑i=1,N riφi (t), i.e. r(t) can

not be expressed as a linear combination of building-block waveforms.If we now for all m ∈ M take a filter with impulse response sm(T − t), let the waveformchannel output r(t) be the input of all these filters and sample the M filter outputs at t = T ,we obtain ∫ ∞

−∞r(α)sm(T − t + α)dα

t=T=∫ ∞

−∞r(α)sm(α)dα = (sm · r). (4.19)

This gives another method to determine (sm ·r) and hence to form an optimum receiver (seefigure 4.6). This receiver is called a direct receiver since the filters are matched directlyto the signals {sm(t), m ∈ M}. We again assume that the signals are non-zero only for0 ≤ t ≤ T .

Note that a direct receiver is usually more expensive than a receiver with filters matched tothe building-block waveforms, since always M ≥ N and in practice even often M ≫ N .The weighting-matrix operations are not needed here however.

4.7 Signal-to-Noise RatioWe have seen in the previous section that a matched-filter receiver is optimum, i.e. minimizesthe expected error probability PE . Not only in this sense is the matched filter optimum, we willshow next that it also maximizes the signal-to-noise ratio. To see what we mean by this, considerthe communication situation shown in figure 4.7. A signal s(t) is assumed to be non-zero only


for 0 ≤ t ≤ T . This signal is observed in additive white noise, i.e. the observer receivesr(t) = s(t)+nw(t). The process Nw(t) is a zero-mean white Gaussian noise process with powerdensity Sw( f ) = N0/2 for all −∞ < f < ∞.

The problem is now e.g. to decide whether the signal s(t) was present in the noise or not(or in a similar setting to decide whether s(t) or −s(t) was seen). Therefore the observer uses alinear time-invariant filter with impulse response h(t) and samples the filter output at time t = T .

For the sampled filter output u(t) at time t = T we can write

u(T ) =∫ ∞

−∞r(T − α)h(α)dα = us(T ) + un(T ), (4.20)

with

us(T )1=

∫ ∞

−∞s(T − α)h(α)dα (4.21)

un(T )1=

∫ ∞

−∞nw(T − α)h(α)dα, (4.22)

where us(T ) is the signal component and un(T ) the noise component in the sampled filter output.

Definition 4.1 We can now define the signal-to-noise ratio as

S/N 1= u2s (T )

E[U 2n (T )]

, (4.23)

i.e. the ratio between signal energy and noise variance.

The noise variance can be expressed as

E[U 2n (T )] = E[

∫ ∞

−∞Nw(T − α)h(α)dα

∫ ∞

−∞Nw(T − β)h(β)dβ]

=∫ ∞

−∞

∫ ∞

−∞E[Nw(T − α)Nw(T − β)]h(α)h(β)dαdβ

= N0

2

∫ ∞

−∞

∫ ∞

−∞δ(β − α)h(α)h(β)dαdβ = N0

2

∫ ∞

−∞h2(α)dα. (4.24)

?

HHH- -�� us(T ) + un(T )

sample at t = T

r(t) = s(t) + nw(t)h(t)

Figure 4.7: A matched filter maximizes S/N .

4.8. EXERCISES 67

RESULT 4.3 If we substitute (4.24) and (4.21) in (4.23) we obtain for the maximum attainablesignal-to-noise ratio

S/N =[∫ ∞

−∞ s(T − α)h(α)dα]2

N02

∫ ∞−∞ h2(α)dα

(∗)≤∫ ∞−∞ s2(T − α)dα

∫ ∞−∞ h2(α)dα

N02

∫ ∞−∞ h2(α)dα

=∫ ∞−∞ s2(T − α)dα

N02

= EsN02

. (4.25)

The inequality (∗) in this derivation comes from Schwarz inequality1 Equality is obtained if andonly if h(t) = Cs(T − t) for some constant C, i.e. if the filter h(t) is matched to the signal s(t).

Note that the maximum signal-to-noise ratio depends only on the energy of the waveforms(t). In this section we have demonstrated a much weaker form of optimality for the matchedfilter than the one that we have obtained in the previous chapter. The matched filter is not onlythe filter that maximizes signal-to-noise ratio, but can be used for optimum detection as well.

4.8 Exercises1. One of two equally likely messages is to be transmitted over an additive white Gaussian

noise channel with SNw( f ) = N0/2 = 1 by means of binary pulse-position modulation.

Specifically

s1(t) = p(t),s2(t) = p(t − 2),

for which the pulse p(t) is shown in figure 4.8.

0 1 2 t

1

2 p(t)

Figure 4.8: A rectangular pulse.

1For two finite-energy waveforms a(t) and b(t) the inequality(∫ ∞

−∞a(t)b(t)dt

)2

≤∫ ∞

−∞a2(t)dt

∫ ∞

−∞b2(t)dt (4.26)

holds. Equality is obtained if and only if b(t) = Ca(t) for some constant C .


(a) Describe (and sketch) an optimum receiver for this case? Express the resulting errorprobability in terms of Q(·).

(b) Give the implementation of an optimum receiver which uses a single linear filterfollowed by a sampler and comparison device. Assume that two samples from thefilter output are fed into the comparison device. Sketch the impulse response of theappropriate filter. What is the output of the filter at both sample moments when thefilter input is s1(t)? What are these outputs for filter input s2(t)?

(c) Calculate the minimum attainable average error probability if

s1(t) = p(t) and s2(t) = p(t − 1).

(Exam Communication Principles, October 6, 2003.)

2. In a communication system based on an additive white Gaussian noise waveform channelsix signals (waveforms) are used. All signals are zero for t < 0 and t ≥ 8. For 0 ≤ t < 8the signals are

s1(t) = 0s2(t) = +2 cos(π t/2)

s3(t) = +2 cos(π t/2) + 2 sin(π t/2)

s4(t) = +2 cos(π t/2) + 4 sin(π t/2)

s5(t) = +4 sin(π t/2)

s6(t) = +2 sin(π t/2)

The messages corresponding to the signals all have probability 1/6. The power spectraldensity of the noise process Nw(t) is N0/2 = 4/9 for all f . The receiver observes thereceived waveform r(t) = sm(t) + nw(t) in the time interval 0 ≤ t < 8.

(a) Determine a set of building-block waveforms for these six signals. Sketch thesebuilding-block waveforms. Show that they are orthonormal over [0, 8). Give thevector representations of all six signals and sketch the resulting signal structure.

(b) Describe for what received vectors r an optimum receiver chooses M̂ = 1, M̂ =2, · · · , and M̂ = 6. Sketch the corresponding decision regions I1, I2, · · · , I6. Givean expression for the error probability PE obtained by an optimum receiver. Use theQ(·)-function.

(c) Sketch and specify a matched-filter implementation of an optimum receiver for thesix signals (all occurring with equal a-priori probability). Use only two matchedfilters.

(Exam Communication Theory, November 18, 2003.)

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