EE303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 1 / 36
Table of Contents 1 Introduction Communication Systems - Block Diagrams Simplified Block Diagrams 2 Communication Signals Classification of Signals Some Signal Parameters Signal Bandwidth Redundancy - Autocorrelation Similarity - Cross Correlation 3 Parseval s and Wiener-Khinchin(W-K) Theorems 4 Woodwards s Notation and Fourier Transform Introduction Examples Fourier Transform Tables 5 Comm Systems: Basic Performance Criteria 6 Additive Noise Additive White Gaussian Noise (AWGN) Bandlimited AWGN "I" and "Q" Noise Components 7 Tail function (or Q-function) for Gaussian Signals Probablity and Probability-Density-Function (pdf) Gaussian pdf and Tail function Tail Function Graph Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 2 / 36
Introduction Communication Systems - Block Diagrams Block Structure of a Digital Comm System Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 3 / 36
Introduction Communication Systems - Block Diagrams Block Structure of a Spread Spectrum Comm System Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 4 / 36
Introduction Communication Systems - Block Diagrams Multi-carrier Comm Systems Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 5 / 36
Introduction Simplified Block Diagrams Simplified Block Diagrams A simplified and general block structure of a Digital Communication System is shown in the following page. it is common practice its quality to be expressed in terms of the accuracy with which the binary digits delivered at the output of the detector/rx (point B) represent the binary digits that were fed into the digital modulator/tx (point B). It is generally taken that it is the fraction of the binary digits that are delivered back in error that is a measure of the quality of the communication system. This fraction, or rate, is referred to as the probability of a bit in error, or, Bit-Error-Rate BER (point B). Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 6 / 36
Introduction Simplified Block Diagrams Digital Communication Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 7 / 36
Introduction Simplified Block Diagrams Digital Transmission of Analogue Signals Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 8 / 36
Introduction Simplified Block Diagrams cont. N.B.: Quality is measured as the Signal-to-Noise (SNR) at point T - known as input SNR (analogue signals degrade as noise level increases) the SNR at point  - known as output SNR Bit-Error-Rate at point B Note that like Wireline and fiber communications wireless communications are also fully digital. It is clear from the previous discussion that signals (representing bits) propagate through the networks. Therefore the following sections are concerned with the main properties and parameters of communication signals. Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 9 / 36
Communication Signals Communication Signals Frequency Domain (spectrum) is very important in Communications Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 10 / 36
Communication Signals Classification of Signals Classifications of Signals According to their description Deterministic Signals Random * describable by mathematical functions * these are unpredictable * cannot be expressed as a function * can be expressed probabilistically Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 11 / 36
Communication Signals Classification of Signals Classifications of Signals According to their periodicity Periodic Non-Periodic N.B.:according to Fourier Series Theorem any periodic waveform can be represented by a sum of sinusoidals having frequencies F 0 = 1 T 0, 2F 0, 3F 0, etc. Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 12 / 36
Communication Signals Classification of Signals Classifications of Signals According to their Energy Energy Signals (Energy< ) Power Signals (Energy= ) N.B.: Energy= g(t)2 dt Signals of finite duration are Energy Signals Periodic Signals are Power Signals Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 13 / 36
Communication Signals Classification of Signals Classifications of Signals According to their Spectrum Low Pass Signals Band Pass Signals Other types Bandwidth=W W =max Freq. Bandwidth=B e.g all-pass, high-pass Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 14 / 36
Communication Signals Classification of Signals Classifications of Signals (cont.) According to their Spectrum (Power Spectral Density, PSD(f)) Low Pass Signals (Bandwidth=W ) Band Pass Signals Bandwidth=B Fractional Bandwidth= B F c Narrow Band Wide Band Ultra Wide Band 0 < B fr < 0.01 0.01 < B fr < 0.25 0.25 < B fr < 2.00 Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 15 / 36
Communication Signals Some Signal Parameters Some Signal Parameters The figures below show the following parameters: peak (Volts) or peak-to-peak Energy (J) (or Power (W)) - over 1Ω resistor rms (Volts) Crest Factor: CF= peak rms (.)2 Power= Energy T Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 16 / 36
Communication Signals Signal Bandwidth Signal Bandwidth Bandwidth of a signal: is the range of the significant frequency components in a signal waveform Examples of message signals (baseband signals) and their bandwidth: television signal bandwidth 5.5MHz speech signal bandwidth 4KHz audio signal bandwidth 8kHz to 20kHz Note that there are various definitions of bandwidth, e.g. 3dB bandwidth, (see B 2 ) or 10dB bandwidth null-to-null bandwidth, (see B 1 = T 2,where T =signal duration). Nyquist (minimum) bandwidth (see B 3 = T 1 ) Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 17 / 36
Communication Signals Redundancy - Autocorrelation Redundancy - Autocorrelation The degree of Redundancy in a signal is provided by its autocorrelation function. For instance the autocorrelation function of a signal is R gg (τ) NB: if τ = fixed then R gg (τ) = a number if τ = variable (i.e. τ) then R gg (τ) = a function of τ Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 18 / 36
Communication Signals Similarity - Cross Correlation Similarity - Cross Correlation The degree of Similarity between two signals is given by their cross-correlation function. For instance the cross-correlation function between two signals g 1 (t) and g 2 (t) is R g1 g 2 (τ) NB: if τ = fixed then R g1 g 2 (τ) = a number if τ = variable (i.e. τ) then R g1 g 2 (τ) = a function of τ Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 19 / 36
Parseval s and Wiener-Khinchin(W-K) Theorems Parseval s and Wiener-Khinchin(W-K) Theorems Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 20 / 36
Parseval s and Wiener-Khinchin(W-K) Theorems "Accumulators" and "Averaging" Devices Definitions Accumulators: t2 Averaging: 1 t 2 t 1 t2 t 1 1 M Examples Accumulators corresponding to Energy Averaging corresponding to Power t 1 M i=1 M i=1 R gg (τ) t2 t 1 E{.} g(t).g(t τ)dt ESD g (f ) =FT{R gg (τ)} R gg (τ) 1 t2 =t 1 +τ t 2 t 1 t 1 R gg (τ) E{g(t).g(t τ)} PSD g (f ) =FT{R gg (τ)} g(t).g(t τ)dt Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 21 / 36
Woodwards s Notation and Fourier Transform Introduction Woodwards s Notation and FT The evaluation of FT, that is FT{g(t)} = G (f ) FT 1 {G (f )} = g(t) g(t). exp( j2πft).dt (1) G (f ). exp(+j2πft).df (2) involves integrating the product of a function and a complex exponential - which can be diffi cult; so tables of useful transformations are frequently used. However, the use of tables is greatly simplified by employing Woodward s notation for certain commonly occurring situations. Main advantage of using Woodward s notation: allows periodic time/frequency functions to be handled with FT rather than Fourier Series. Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 22 / 36
Woodwards s Notation and Fourier Transform Introduction cont. 1. rect{t} { 1 if t < 1 2 0 otherwise 2. sinc{t} sin(πt) πt 3. Λ{t} { 1 t if 0 t 1 1 + t if 1 t 0 Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 23 / 36
Woodwards s Notation and Fourier Transform Introduction cont. 4. rep T {g(t)} + g(t nt ) n :... 2, 2, 1, 0, 1,... n= 5. comb T {g(t)} + g(nt ).δ(t nt ) n= also known as sampling function Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 24 / 36
Woodwards s Notation and Fourier Transform Examples Examples we can generate any desired "rect" function by scaling and shifting - see for instance the following table shifting scaling shifting+scaling { } { } t t τ g(t) =rect{t τ} g(t) =rect g(t) =rect T T 1 2 t τ 1 2 1 2 t T 1 2 1 2 t τ T 1 2 τ 1 2 t τ + 1 2 T 2 t T 2 τ T 2 t τ + T 2 Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 25 / 36
Woodwards s Notation and Fourier Transform Examples Effects of Scaling as T FT becomes narrower and amplitude rises δ-function at 0 frequency when T Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 26 / 36
Woodwards s Notation and Fourier Transform Fourier Transform Tables Fourier Transform Tables Description Function Transformation 1 Definition g(t) G (f ) = g(t).e j2πft dt 2 Scaling g ( ) t T T.G (ft ) 3 Time shift g(t T ) G (f ).e j2πft 4 Frequency shift g(t).e j2πft G (f F ) 5 Complex conjugate g (t) G ( f ) 6 Temporal derivative d n dt g(t) n (j2πf )n.g (f ) 7 Spectral derivative ( j2πt) n.g(t) d n df G (f ) n 8 Reciprocity G (t) g( f ) 9 Linearity A.g(t) + B.h(t) A.G (f ) + B.H(f ) 10 Multiplication g(t).h(t) G (f ) H(f ) 11 Convolution g(t) h(t) G (f ).H(f ) 12 Delta function δ(t) 1 13 Constant 1 δ(f ) { 1 if t < 1 14 Rectangular function rect{t} 2 sinc{f } sin(πf ) 0 otherwise πf 15 Sinc function sinc(t) rect{f } { 1 t > 0 16 Unit step function u(t) 0 t < 0 17 Signum function sgn(t) 18 decaying exp (two-sided) 19 decaying exp (one-sided) { 1 t > 0 1 t < 0 1 2 δ(f ) j 2πf j πf e t 2 1+(2πf ) 2 e t.u(t) 1 j2πf 1+(2πf ) 2 20 Gaussian function e πt2 e πf 2 { 1 t if 0 t 1 21 Lambda function Λ{t} sinc 1 + t if 1 t 0 2 {f } 22 Repeated function rept {g(t)} = g(t) rept {δ(t)} 1 T comb 1T {G (f )} 23 Sampled function combt {g(t)} = g(t).rept {δ(t)} 1 T rep 1 {G (f )} T Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 27 / 36
Comm Systems: Basic Performance Criteria Basic Performance Criteria SNR in = Power of signal at T {(βs(t)) Power of noise at T = E 2} E {n(t) 2 } = β2 P s N 0 B }{{} P n p e = BER at point B (4) SNR out = (3) Power of signal at  Power of noise at  = f{p }{{ e} (5) } denotes: a function of p e Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 28 / 36
Additive Noise Additive Noise types of channel signals s(t), r(t), n(t): bandpass n i (t) = AWGN: allpass Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 29 / 36
Additive Noise Additive White Gaussian Noise (AWGN) n i (t) it is a random all-pass signal its Power Spectral Density is "White" i.e. "flat". That is, PSD ni (f ) = N 0 2 (6) its amplitude probability density ( function is Gaussian ) its Autocorrelation function i.e. FT 1 {PSD(f )} is: R ni n i (τ) = N 0 2 δ(τ) (7) Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 30 / 36
Additive Noise Bandlimited AWGN Bandlimited AWGN n(t) it is a random Band-Pass signal of bandwidth B (equal to the channel bandwidth) its Power Spectral Density is "bandmimited White". That is, PSD ni (f ) = N 0 2 Its power is: ( { } { }) f + Fc f Fc rect + rect B B P n = σ 2 n = PSD ni (f ).df = N 0 2 B 2 P n = N 0 B (9) (8) Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 31 / 36
Additive Noise Bandlimited AWGN more on n(t): its amplitude probability density function is Gaussian It is also known as bandlimited-awgn It can be written as follows: where pdf n = N(0, σ 2 n = N 0 B) (10) n(t) = n c (t) cos(2πf c t) n s (t) sin(2πf c t) (11) = } nc 2 (t) + ns 2 (t) cos(2πf c t + φ n (t)) {{} (12) r n (t) n c (t) and n s (t) are random signals - with pdf=gaussian distribution r n (t) is a random signal - with pdf=rayleigh distribution φ n (t) is a random signal - with pdf=uniform distribution: [0, 2π]) N.B.: all the above are low pass signals & appear at Rx s o/p Equ. 11 is known as Quadrature Noise Representation. Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 32 / 36
Additive Noise "I" and "Q" Noise Components "I" and "Q" Noise Components n c (t) (i.e. "I") and n s (t) (i.e. "Q") their Power Spectral Densities are: their power are: { } f PSD nc (f ) = PSD ns (f ) = N 0 rect B P nc = σ 2 n c = PSD nc (f ).df = N 0 B (13) P nc = P ns = P n = N 0 B (14) Amplitude probability density functions: Gaussian, are uncorrelated i.e. E {n c (t).n s (t)} = 0 pdf nc = pdf ns = N(0, N 0 B) (15) Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 33 / 36
Tail function (or Q-function) for Gaussian Signals Probablity and Probability-Density-Function (pdf) Tail function (or Q-function) for Gaussian Signals Probablity and Probability-Density-Function (pdf) Consider a random signal x(t) with a known amplitude probability density function pdf x (x) - not necessarily Gaussian. Then the probability that the amplitude of x(t) is greater than A Volts (say) is given as follows: Pr(x(t) > A) = A pdf x (x).dx (16) e.g. if A = 3V Pr(x(t) > 3V ) = 3 pdf x (x).dx = highlighted area Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 34 / 36
Tail function (or Q-function) for Gaussian Signals Gaussian pdf and Tail function Gaussian pdf and Tail function If pdf x (x) = Gaussian of mean µ x and standard deviation σ x (notation used: pdf x (x) = N(µ x, σ 2 x ), then the above area is defined as the Tail-function (or Q-function) e.g. Pr(x(t) > A) = A pdf x (x).dx T{ A µ x σ x } (17) if pdf x (x) = N(1, 4) - i.e. µ x = 0, σ x = 2 - and A = 3V then Pr(x(t) > 3V ) = 3 pdf x (x).dx T{ 3 1 if pdf x (x) = N(0, 1) and A = 3V then Pr(x(t) > 3V ) = 3 pdf x (x).dx T{3} The Tail function graph is given in the next page 2 }=T{1} Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 35 / 36
Tail function (or Q-function) for Gaussian Signals Tail Function Graph Prof. A. Manikas (Imperial College) EE303: Introductory Concepts v.17 36 / 36