Communication Channels

Professor A. Manikas

Imperial College London

EE303 - Communication Systems An Overview of Fundamentals

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 1 / 48 Table of Contents 1 Introduction 2 Continuous Channels 3 Discrete Channels 4 Converting a Continuous to a Discrete Channel 5 More on Discrete Channels Backward transition Matrix Joint transition Probability Matrix 6 Measure of Information at the Output of a Channel Mutual Information of a Channel Equivocation & Mutual Information of a Discrete Channel 7 Capacity of a Channel Shannos’sCapacity Theorem Capacity of AWGN Channels Capacity of non-Gaussian Channels Shannon’sChannel Capacity Theorem based on Continuous Channel Parameters 8 Bandwidth and Channel Symbol Rate 9 Criteria and Limits of DCS Digital Communications - Introduction ENERGY UTILIZATION EFFICIENCY (EUE) Bandwidth Utilisation Eﬃ ciency (BUE) Visual Comparison of Comm Systems Theoretical Limits on the Performance of Dig. Comm. Systems 10 Other Comparison-Parameters 11 Appendix-A: SNR at the output of an Ideal Comm System Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 2 / 48 Introduction Introduction

With reference to the following block structure of a Dig. Comm. System (DCS), this topic is concerned with the basics of both continuous and discrete communication channels.

Block structure of a DCS

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 3 / 48 Introduction Just as with sources, communication channels are either

I discrete channels, or I continuous channels 1 wireless channels (in this case the whole DCS is known as a Wireless DCS) 2 wireline channels (in this case the whole DCS is known as a Wireline DCS) Note that a continuous channel is converted into (becomes) a discrete channel when a digital modulator is used to feed the channel and a digital demodulator provides the channel output. Examples of channels - with reference to DCS shown in previous page,

I discrete channels:

F input: A2 - output: A2 (alphabet: levels of quantiser - Volts) F input: B2 - output: B2 (alphabet: binary digits or binary codewords) b I continuous channels: b F input: A1 - output: A1, (Volts) - continuous channel (baseband) F input: T, - output: T (Volts) - continuous channel (baseband), F input: T1 - output: T1b (Volts) - continuous channel (bandpass). b Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 4 / 48 b Continuous Channels Continuous Channels A continuous communication channel (which can be regarded as an analogue channel) is described by I an input ensemble (s(t), pdfs (s)) and PSDs (f ) I an output ensemble, (r(t), pdfr (r)) I the channel noise (AWGN) ni (t) and β, I the channel bandwidth B and channel capacity C.

⇐⇒

Prof. A.types Manikas of (Imperial channel College) signalsEE303: Communication Channels 2011 5 / 48 I s(t), r(t), n(t): bandpass I ni (t) = AWGN: allpass Discrete Channels Discrete Channels A discrete communication channel has a discrete input and a discrete output where

I the symbols applied to the channel input for transmission are drawn from a ﬁnite alphabet, described by an input ensemble (X , p) while I the symbols appearing at the channel output are also drawn from a ﬁnite alphabet, which is described by an output ensemble (Y , q) I the channel transition probability matrix F.

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 6 / 48 Discrete Channels

In many situations the input and output alphabets X and Y are identical but in the general case these are diﬀerent. Instead of using X and Y , it is common practice to use the symbols H and D and thus deﬁne the two alphabets and the associated probabilities as

,p1 ,p2 ,pM T input: H = H1, H2, ..., HM p = [Pr(H1), Pr(H2), ..., Pr(HM )] { } z ,}|q1 { z ,}|q2 { z }|,qK { T output: D = D1, D2, ..., DM q = [Pr(D1), Pr(D2), ..., Pr(DK )] { } z }| { z }| { z }| { where pm abbreviates the probability Pr(Hm ) that the symbol Hm may appear at the input while qk abbreviates the probability Pr(Dk ) that the symbol Dk may appear at the output of the channel.

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 7 / 48 Discrete Channels The probabilistic relationship between input symbols H and output symbols D is described by the so-called channel transition probability matrix F, which is deﬁned as follows:

Pr(Dk Hm ) denotes the probability that symbol Dk D will appear | ∈ at the channel output, given that Hm H was applied to the input. ∈ The input ensemble H, p , the output ensemble D, q and the matrix F fully describe the functional properties of the channel. The following expression describes the relationship between q and p

q = F.p (2)

Note that in a noiseless channel

D = H (3) q = p

i.e the matrix F is an identity matrix

F = IM (4)

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 9 / 48 Converting a Continuous to a Discrete Channel Converting a Continuous to a Discrete Channel A continuous channel is converted into (becomes) a discrete channel when a digital modulator is used to feed the channel and a digital demodulator provides the channel output. A digital modulator is described by M diﬀerent channel symbols . These channel symbols are ENERGY SIGNALS of duration Tcs . Digital Modulator:

Digital Demodulator:

If M = 2 Binary Digital Modulator Binary Comm. System If M > 2 ⇒M-ary Digital Modulator ⇒M-ary Comm. System ⇒ ⇒ Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 10 / 48 Converting a Continuous to a Discrete Channel

..01..101..00.. T T 0..00 st( )= : H ,Pr(H ) cs cs 1 t 1 1 st( )= t 0..01 st( )= : H ,Pr(H ) 2 t 2 2 Tcs Tcs t 1..11 stM( )= : HMM , Pr(H ) st() Channel

..00..101..10.. rt()=()+n() st t 0..00 st( )= : D , Pr(D ) 1 t 1 1 Tcs Tcs D Detector rt( )= 0..01 st2( )= : 2D 2 , Pr(D ) with a t Decision Device t

t Tcs Tcs 1..11 stM( )= : DMM , Pr(D ) (Decision Rule)

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 11 / 48 More on Discrete Channels Backward transition Matrix More on Discrete Channels Backward transition Matrix

There are also occassions where we get/observe the output of a channel and then, based on this knowledge, we refer to the input .

In this case we may use the concept of an imaginary "backward " channel and its associated transition matrix, known as backward transition matrix deﬁned as follows:

T Pr(H1 D1), Pr(H1 D2), ..., Pr(H1 DK ) Pr(H |D ), Pr(H |D ), ..., Pr(H |D ) B = 2 1 2 2 2 K (5)  ...,| ...,| ..., ...|  Pr(H D ), Pr(H D ), ..., Pr(H D )  M 1 M 2 M K   | | | 

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 12 / 48 More on Discrete Channels Joint transition Probability Matrix Joint transition Probability Matrix The joint probabilistic relationship between input channel symbols H = H1, H2, ..., HM and output channel { } symbols D = D1, D2, ..., DM , is described by{ the so-called joint-probability} matrix, T Pr(H1, D1), Pr(H1, D2), ..., Pr(H1, DK ) Pr(H , D ), Pr(H , D ), ..., Pr(H , D ) J 2 1 2 2 2 K (6) ,  ..., ..., ..., ...  Pr(H , D ), Pr(H , D ), ..., Pr(H , D )  M 1 M 2 M K  J is related to the forward transition probabilities of a channel with the following expression (compact form of Bayes’Theorem):

p1 0 ... 0 0 p ... 0 J = F. 2 = F.diag(p) (7) ......   0 0 ... p   M    ,diag(p) Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 13 / 48 | {z } More on Discrete Channels Joint transition Probability Matrix

Note: This is equivalent to a new (joint) source having alphabet

(H1, D1) , (H1, D2) , ..., (HM , DK ) { } and ensemble (joint ensemble) deﬁned as follows

(H1, D1), Pr(H1, D1) (H1, D2), Pr(H1, D2)  ...  (H D, J) =   (8) ×  (Hm, Dk ), Pr(Hm, Dk )     ...   (HM , DK ), Pr(HM , DK )        = (Hm, Dk ), Pr(Hm, Dk ) , mk : 1 m M, 1 k K   ∀ ≤ ≤ ≤ ≤   =J   km     | {z }  Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 14 / 48 Measure of Information at the Output of a Channel Measure of Information at the Output of a Channel

In general three measures of information are of main interest:

1 the Entropy of a Source - in (info) bits per source symbol

2 the Mutual Entropy (or Multual Information) of a Channel , in (info) bits per channel symbol

3 the Discrimination of a Sink

Next we will focus on the Mutual Information of a Channel Hmut

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 15 / 48 Measure of Information at the Output of a Channel Mutual Information of a Channel Mutual Information of a Channel

The mutual information measures the amount of information that the output of the channel (i.e. received message) gives about the input to the channel (transmitted message).

That is, when symbols or signals are transmitted over a noisy communication channel, information is received. The amount of information received is given by the mutual information,

Hmut = 0 (9)

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 16 / 48 Measure of Information at the Output of a Channel Mutual Information of a Channel

M K qk Hmut , Hmut (p, F) = ∑ ∑ Fkm.pm log2 (10) − Fkm m=1 k=1 M K pm.qk = ∑ ∑ Jkm log2 (11) − Jkm m=1 k=1

T T bits = 1K J log2 F.p.p ∅J  1M − symbol  K hM matrix i  ×    (12) | {z } where

1M = a column vector of M ones , ∅ = Hadamard operators (mult. and div.)

Note that 1T A1 = adds all elements of A (13)

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 17 / 48 Measure of Information at the Output of a Channel Equivocation & Mutual Information of a Discrete Channel Equivocation & Mutual Information of a Discrete Channel Consider a discrete source (H, p) followed by a discrete channel, as shown below The average amount of information gained (or uncertainty removed) about the H source (channel input) by observing the outcome of the D source (channel output), is given by the conditional entropy HH D which is deﬁned as follows: | M K Jkm HH D , HH D (J) = ∑ ∑ Jkm. log2 (14) | | − qk m=1 k=1

B   T 1 bits = 1 J log diag (q)− J 1 − K  2   M symbol    z }| {      KM matrix   ×    (15) | {z } Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 18 / 48 Measure of Information at the Output of a Channel Equivocation & Mutual Information of a Discrete Channel

A similar expression can be also given for the average information gained about the channel output D by observing the channel input H, i.e.

M K Jkm HD H , HD H (J) = ∑ ∑ Jkm. log2 (16) | | − pm m=1 k=1

F   T 1 bits = 1 J log J.diag (p)− 1 − K  2   M symbol    z }| {      KM matrix   ×    (17) | {z }

The conditional entropy HH D is also known as equivocation and it is the entropy of the noise| or, otherwise, the uncertainty in the input of the channel from the receiver’spoint of view.

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 19 / 48 Measure of Information at the Output of a Channel Equivocation & Mutual Information of a Discrete Channel

Notes

1 for a noiseless channel: HH D = 0 (18) |

2 For a discrete memoryless channel,

Hmut , Hmut (p, F) = HH HH D (19) − | = HD HD H (20) − |

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 20 / 48 Capacity of a Channel Shannos’sCapacity Theorem Capacity of a Channel Shannon’s Capacity Theorem

There is a theoretical upper limit to the performance of a specified digital communication system with the upper limit depending on the actual system specified. However, in addition to the specific upper limit associated with each system, there is an overall upper limit to the performance which no digital communication system, and in fact no communication system at all, can exceed. This bound (limit) is important since it provides the performance level against which all other systems can be compared. The closer a system comes, performance wise, to the upper limit the better.

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 21 / 48 Capacity of a Channel Shannos’sCapacity Theorem The theoretical upper limit was given by Shannon (1948) as an upper bound to the maximum rate at which information can be transmitted over a communication channel. This rate is called channel capacity and is denoted by the symbol C. Shannon’scapacity theorem states: bits C , max (Hmut ) symbol (21) or bits C rcs max (Hmut ) (22) , × sec where rcs denotes the channel-symbol rate (in channel-symbols per sec) with 1 rcs = (23) Tcs r B cs (24) ≥ 2 i.e. if Hmut (p, F) is maximised with respect to the input probabilities p, then it becomes equal to C, the channel capacity (in bits/symbol)

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 22 / 48 Capacity of a Channel Capacity of AWGN Channels Capacity of AWGN Channels

In the case of a continuous channel corrupted by additive white Gaussian noise the capacity is given by 1 C = log (1 + SNR ) bits (25) 2 2 in symbol or (26) bits C = B log2 (1 + SNRin) sec (27) where

B = baseband band width of channel Ps SNRin = Pn

Ps = Power of the signal at point T

Pn = Power of the noise at point T = N0B b

Prof. A. Manikas (Imperial College) EE303: Communication Channels b 2011 23 / 48 Capacity of a Channel Capacity of non-Gaussian Channels Capacity of non-Gaussian Channels If the pdf of the noise is arbitrary (non-Gaussian) then it is very diﬃ cult to estimate the capacity . However, it can be proved [Shannon 1948] that in this case the capacity is bounded as follows:

Ps + Nn Ps + Pn bits B log2 C B log2 sec (28) Nn ≤ ≤ Nn where

Ps : is the average received signal power,

Nn : is the entropy power of the noise, and

Pn : is the power of the noise

Equation 28 is important in that it can be used to provide bounds for any kind of channel.

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 24 / 48 Shannon’sChannel Capacity Theorem based on Continuous Capacity of a Channel Channel Parameters Shannon’sChannel Capacity Theorem based on Continuous Channel Parameters Consider a time-continuous channel which comprises of a linear time invariant ﬁlter with transfer function H(f ), the output of which is corrupted by an additive zero mean stationary noise n(t) of PSDn(f ). I if the average power of the channel input signal is constraint to be Ps , then ∞ PSDn(f ) Ps = max 0, θ .df (29) − H(f ) 2 Z∞ ( ) − | | ∞ 1 θ. H(f ) 2 and C max 0, log2 | | .df (30) ≥ 2 PSDn(f ) Z∞ ( !) − with the equality holding if the noise is Gaussian. N0 I Further, if the channel noise is white Gaussian with PSDn(f ) = 2 then Equation 30 simpliﬁes to the well known result bits C = B log2 (1 + SNRin) sec (31)

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 25 / 48 Bandwidth and Channel Symbol Rate Bandwidth and Channel Symbol Rate

The following expression are given without any proof: channel symbol rate Baseband Bandwidth (32) ≥ 2 channel symbol rate Bandpass Bandwidth 2 (33) ≥ 2 × The equality is known as Nyquist Bandwidth. In this course, except if it is deﬁned otherwise,

I the word "bandwidth" will mean "Nyquist bandwidth" I the carrier will be ignored and thus "bandwidth" by default will refer to "baseband bandwdith"

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 26 / 48 Criteria and Limits of DCS Digital Communications - Introduction Critteria and Limits of DCS Digital Communications - Introduction

Digital Communications provide excellent message-reproduction and greatest Energy (EUE) and Bandwidth (BUE) Utilization Eﬃ ciency through eﬀective employment of two fundamental techniques:

I source compression coding (to reduce the transmission rate for a given degree of ﬁdelity) I error control coding and digital modulation (to reduce the SNR and bandwidth requirements) With reference to the general structure of a DCS given in the next page,

I the source compression coding is implemented by the blocks "Source Encoder" and "Source Decoder" I the error control coding is implemended by the "Discrete Channel Endoder", "Interleaver", "DeInterleaver" and "Discrete Channel Decoder".

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 27 / 48 Criteria and Limits of DCS Digital Communications - Introduction

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 28 / 48 Criteria and Limits of DCS Digital Communications - Introduction Let us focus on the Discrete Channel: We have seen that a digital modulator is described by M = 2γcs diﬀerent channel symbols which are ENERGY SIGNALS of duration Tcs .

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 29 / 48 Criteria and Limits of DCS ENERGY UTILIZATION EFFICIENCY (EUE) Energy Utilisation Effi ciency (EUE) The parameter EUE is a measure of how effi ciently the system utilises the available energy in order to transmit information in the presence of additive white Gaussian noise of double-sided power spectral density PSDn(f ) = N0/2 and it is defined as follows:

Eb EUE , (34) N0 Note that EUE is directly related to the received signal power. It will be appreciated of course that this is, in turn, directly related to the transmitted power by the attenuation factor introduced by the channel. Clearly, a question of major importance is how large EUE needs to be in order to achieve communication at some speciﬁc bit error probability pe . Obviously the smaller EUE to achieve a speciﬁed error probability the better .

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 30 / 48 Criteria and Limits of DCS Bandwidth Utilisation Eﬃ ciency (BUE) Bandwidth Utilisation Eﬃ ciency (BUE)

The BUE measures how eﬃ ciently the system utilises the bandwidth B available to send information and it is deﬁned as follows: B BUE , (35) rb

where rb denotes the bit rate. Speciﬁcally, the BUE indicates how much bandwidth is being used per transmitted information bit and hence, for a given level of performance, the smaller BUE the better since this means that less bandwidth is being used to achieve a given rate of data transmission. N.B.: r signaling speed b = BUE 1 (36) , B −

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 31 / 48 Criteria and Limits of DCS Visual Comparison of Comm Systems Visual Comparison of Comm Systems By using EUE and BUE the SNRin can be expressed as follows

Eb Eb Ps Tb Eb Eb N0 EUE SNRin = = = = 1 = = (37) Pn N0BTb N B B BUE N0B 0 rb rb By determining the EUE and BUE of any particular system, that system can be represented as a point in the plane (EUE,BUE). It is desirable for this point to be as close to the origin as possible

EUE bits C = B log2 1 + BUE sec (38) bits EUE sec C /B = log2 1 + BUE Hz (39)

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 32 / 48 Criteria and Limits of DCS Visual Comparison of Comm Systems

N.B.:

I a line from origin represents those points (systems) in the plane for which the SNRin=constant I By comparing points representing one system with those representing another VISUAL COMPARISON ! ⇒

I It can be observed that

F CS1 beter than CS2 which is better than CS3 F CS2 and CS3 have the same SNRin

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 33 / 48 Criteria and Limits of DCS Theoretical Limits on the Performance of Dig. Comm. Systems Theoretical Limits on the Performance of Dig. Comm. Systems

We have seen that the capacity of a white Gaussian channel of bandwidth B is bits C = B log2 (1 + SNRin) sec (40) Please don’tforget that the above equation refers to bandlimited white-noise channel with a constraint on the average transmitted power. Question: if B = (and in particluar if B = ∞) then C =? Answer : ↑

I From the capacity-equation (Equ 40) it can be seen that B = C ↑ ⇒ ↑ I However, when B tends to ∞ then

Ps C∞ = 1.44 (41) N0

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 34 / 48 Criteria and Limits of DCS Theoretical Limits on the Performance of Dig. Comm. Systems

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 35 / 48 Criteria and Limits of DCS Theoretical Limits on the Performance of Dig. Comm. Systems

LIMIT-1 : limit on bit rate

I when binary information is transmitted in the channel, rb should be limited as follows: r C (42) b ≤ I ideal case: rb = C (43) LIMIT-2 : limit on EUE

I the best Energy Eﬃ ciency is EUE=0.693. This is the ultimate limit below which no physical channel can transmit without errors i.e EUE 0.693 ≥ LIMIT-3 : Shannon’sthreshold channel capacity curve

I This is the curve EUE=f BUE for a bit rate rb equal to its maximum value, { } i.e. 2BUE 1 rb = C EUE = −1 (44) ⇒ BUE−

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 36 / 48 Criteria and Limits of DCS Theoretical Limits on the Performance of Dig. Comm. Systems

Plot of Equation 44:

No physical realizable CS could occupy a point in the plane (EUE,BUE) lying below this theoretical channel capacity curve .

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 37 / 48 Criteria and Limits of DCS Theoretical Limits on the Performance of Dig. Comm. Systems

EUEdata EUEinf SNRin = = (45) BUEdata BUEinf data rate : rb,data = rcs log M (46) × 2 info rate : rb,info = rcs Hmut (47) × Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 38 / 48 Criteria and Limits of DCS Theoretical Limits on the Performance of Dig. Comm. Systems

Point B always should be above the Shannon’sthr. capacity curve. Point A may be above or below the Shannon’sthr. capacity curve. The following table gives the corresponding equivalent parameters between Analogue and Digital Comm Systems which can be used to place on the (EUE,BUE) not only digital but also analogue communication systems

Digital CS Analogue CS

Eb Ps Ps EUE= = SNRin-mb = N0 N0rb ⇔ N0.Fg rb Fg B ⇔ B BUE= β , rb ⇔ Fg where:

I Fg denotes the maximum frequency of the message signal g(t), i.e. it represents the bandwidth of the message I β is known as "bandwidth expansion factor", e.g. SSB: β = 1;AM: β = 2

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 39 / 48 Criteria and Limits of DCS Theoretical Limits on the Performance of Dig. Comm. Systems

Comparison of various Digital and Analogue CS are shown below (for a ﬁxed SNRout )

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 40 / 48 Other Comparison-Parameters Other Comparison-Parameters

SPECTRAL CHARACTERISTICS of transmitted signal (rate at which spectrum falls oﬀ). INTERFERENCE RESISTANCE

I it may be necessary to increase (EUE,BUE) in order to increase interf. resistance FADING

I Fading problem pe = BUE↑= I Note that, if ↑ then Fading= EUE = ↓ ↓ DELAY DISTORTION

I Try to avoid this problem by selecting appropriate signals COST and COMPLEXITY

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 41 / 48 Appendix-A: SNR at the output of an Ideal Comm System Appendix: SNR at the output of an Ideal Comm System

In this section a so-called ideal system of communication will be considered and it will be shown that bandwidth can be exchanged for signal-to-noise performance.

The ideal system forms a benchmark against which other communication systems can be compared. An ideal system has been deﬁned as one that transmits data at a bit rate r = C (48)

where C is the channel capacity i.e. C = B log2 (1 + SNRin) bits/sec

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 42 / 48 Appendix-A: SNR at the output of an Ideal Comm System

Furthermore we have seen that for an ideal communication system:

SNR EUE = in lim EUE = 0.693 (49) log (1 + SNRin) ⇒ SNRin 0 2 →

1 BUE = EUE (50) log2(1 + BUE ) 1 2BUE− 1 EUE = − lim EUE = 0.693 (51) ⇒ BUE 1 ⇒ BUE ∞ − →

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 43 / 48 Appendix-A: SNR at the output of an Ideal Comm System Block Diagram of an Ideal Communication System:

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 44 / 48 Appendix-A: SNR at the output of an Ideal Comm System

The previous ﬁgure shows the elements of a basic ideal communication system.

An input analogue message signal g(t), which is of bandwidth Fg , is applied to a signal mapping unit which, in response to g(t), produces an analogue signal s(t) of bandwidth B and this signal is transmitted over an analogue channel having a similar bandwidth, B.

I The channel is corrupted by additive white Gaussian noise of double sided power spectral density N0/2 which is bandlimited to the channel bandwidth B. I Let the signal-to-noise ratio at the input of the receiver be SNRin. I Assume further that the received signal, plus noise, is then fed to a detector having a bandwidth Fg , equal to the message bandwidth. I Let the signal-to-noise ratio at the output of the detector be SNRout.

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 45 / 48 Appendix-A: SNR at the output of an Ideal Comm System Now the capacity of the analogue transmission system (channel) is

C = B log2 (1 + SNRin) bits/s (52) Also, the "mapping-unit/channel/detector" can be regarded as a channel having a signal-to-noise ratio SNRout and hence it too can be regarded as an AWGN channel and its capacity is given by

C 0 = Fg log2 (1 + SNRout ) bits/s (53) If, in order to avoid information loss (ideal case), the capacities are set equal then it can be seen, after simple mathematical manipulation, that

B F C 0 = C ... SNRout,ideal = (1 + SNRin) g 1 (54) ⇒ ⇒ − SNRin mb β = (1 + − ) 1 (55) β − where β is the bandwidth expansion factor.

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 46 / 48 Appendix-A: SNR at the output of an Ideal Comm System The above expression is fundamentally important since it shows that the overall system performance, SNRout , can be improved by using more channel bandwidth. The ﬁgure below shows, as a function of the bandwidth expansion factor β, typical curves of SNRout versus SNRin mb for the ideal communication system. −

Note that all other known communication systems should be compared with this optimum performance provided by Equation .

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 47 / 48 Appendix-A: SNR at the output of an Ideal Comm System

From the previous ﬁgure it can be seen that:

I if SNRin mb is small then on increasing β the eﬀect on the SNRout is − small (i.e. very little increase in the SNRout is obtained). I If, however, SNRin mb is large then a small increase in the bandwidth − expansion factor results in a large increase in the SNRout . I A practical consequence of this is that if the SNRin mb is small then there is little to be gained from using more channel− bandwidth.

Case-1:SNRin-mb=small Case-2:SNRin-mb=large=10 SNRin-mb=1 SNRin-mb=10 β = 1 SNRout =1 SNRout =10 β = 2 SNRout =1.25 SNRout =35 β = 100 SNRout =1.7048 SNRout =13780

Prof. A. Manikas (Imperial College) EE303: Communication Channels 2011 48 / 48