Analysis and Design of Telecommunications Systems: prerequisite Marc Van Droogenbroeck Abstract This document is the prerequisite for Analysis and Design of Telecommunications Systems taught at the University of Liège. It is intended to be a reminder of some basic knowledge. For a textbook, I recommend: Digital and Analog Communication Systems, by L. Couch, Prentice Hall [1]. 1 Contents 1 Fourier transform and spectrum analysis 3 1.1 Properties . .3 1.2 Specic signals . .4 1.3 Fourier transform pairs . .5 2 Signals 6 2.1 Continuous-time and discrete-time signals . .6 2.2 Analog and digital signals . .6 2.3 Energy and power . .6 2.4 Deterministic vs stochastic tools . .7 2.5 Decibel . .8 2.6 Digitization of analog signals . .8 2.6.1 Sampling theorem . .8 2.6.2 Impulse sampling . .9 3 Linear systems 10 3.1 Linear time-invariant systems . 10 3.2 Impulse response and transfer function . 10 3.3 Distortionless transmission . 10 4 Random variables and stochastic processes 11 4.1 Gaussian random variable . 11 4.2 Stochastic processes . 11 4.2.1 Stationarity . 11 4.2.2 Power spectral density and linear systems (= ltering) . 13 4.2.3 Noise and white noise . 13 5 Line coding and spectra 15 5.1 Line coding . 15 5.2 General formula for the power spectral density of baseband digital signals . 15 6 Power budget link for radio transmissions 17 7 Information theory 18 7.1 Channel capacity . 18 7.2 On the importance of the Eb ratio for digital transmissions . 18 N0 2 1 Fourier transform and spectrum analysis Denition 1.[ Fourier transform] Let g(t) be a deterministic signal (also called a waveform), the Fourier transform of g(t), denoted G, is Z +1 G(f) = g(t)e−2πjftdt (1.1) −∞ where is the frequency parameter with units in Hertz, (that is 1 ). It is also related to angular f Hz s pulsation ! by ! = 2πf. Note that f is the parameter of the Fourier transform G(f), that is also called (two-sided) spectrum of g(t), because it contains both positive and negative components. Denition 2. [Inverse Fourier transform] The time waveform g(t) is obtained by taking the inverse Fourier transform of G(f), dened as follows Z +1 g(t) = G(f)e2πjftdt (1.2) −∞ The functions g(t) and G(f) are said to constitute a Fourier transform pair, because they are two representations of a same signal. 1.1 Properties Proposition 3. [Spectral symmetry of real signals] If g(t) is real, then G(−f) = G∗(f) (1.3) where ()∗ denotes the conjugate operation. A consequence is that the negative components can be obtained from the positives ones. Therefore, the positive components are sucient to describe the original waveform g(t). Proposition 4.[ Rayleigh] The total energy in the time domain and the frequency domain are equal: Z +1 Z +1 E = jg(t)j2 dt = kG(f)k2 df (1.4) −∞ −∞ After integration, kG(f)k2 provides the total energy of the signal in Joule, J. Therefore, it is sometimes named energy spectral density. Proposition 5. [Linearity] If g(t) = c1g1(t) + c2g2(t) then G(f) = c1G1(f) + c2G2(f). This property of linearity is essential in telecommunications, because most systems (channels, lters, etc) are linear. The consequence is that any linear system can only modify the spectral content a signal, but is incapable to add new frequencies. This property also highlights that a spectrum analysis is adequate for dealing with linear systems. To the contrary, it is more dicult to analyze a non-linear system (for example a squaring operator, g2(t)) in terms of frequencies. 3 Operation Function Fourier transform Conjugate symmetry g(t) is real G(f) = G∗(−f) Linearity c1g1(t) + c2g2(t) c1G1(f) + c2G2(f) Time scaling 1 f g(at) jaj G a −2πjft0 Time shift (delay) g(t − t0) G(f) e Convolution R +1 g(t) ⊗ h(t) = −∞ g(τ)h(t − τ) dτ G(f)H(f) 2πjfct Frequency shift g(t) e G (f − fc) Modulation G(f−fc)+G(f+fc) g(t) cos (2πfct) 2 Temporal derivation d dt g(t) 2πjf G(f) Temporal integration R t 1 −∞ g(τ) dτ 2πjf G(f) Table 1: Main properties of the Fourier transform. 1.2 Specic signals Some waveforms play a specic role in communications. 1. The Dirac delta function, denoted δ(t), is dened by 1; x = 0 δ(t) = (1.5) 0 x 6= 0 and Z +1 δ(t) dt = 1 (1.6) −∞ The Dirac delta function is not a true function, so it is said to be a singular function. However, it is considered as a function in the more general framework of the theory of distributions. Note that g(t) ⊗ δ(t) = g(t) (1.7) According to equation 1.6, Z +1 δ(t)e−2πjft = e0 = 1 (1.8) −∞ As a consequence, 1 is the Fourier transform of δ(t). Likewise, δ(f) is the Fourier transform of 1. This explains the central role of the Dirac delta function in signal processing and communications. The Dirac delta function is also called the unit impulse function. 2. Rectangular pulse 1 1 1 − 2 < t < 2 rect(t) = 1 (1.9) 0 jtj > 2 The rectangular pulse is the most common shape form for the representation of digital signals. 3. Step function (Heaviside function) 1 t > 0 u(t) = (1.10) 0 t < 0 4 4. Sign function 1 t > 0 sign(t) = (1.11) −1 t < 0 5. Sinc function sin(πt) sinc(t) = (1.12) πt 1.3 Fourier transform pairs Temporal waveform Fourier transform δ(t) 1 1 δ(f) 1 cos(2πfct) 2 [δ(f − fc) + δ(f + fc)] 1 sin(2πfct) 2j [δ(f − fc) − δ(f + fc)] rect t T T sinc(fT ) ) 1 rect f sinc(2W t 2W 2W sign 1 (t) πjf 1 1 u(t) 2 δ(f) + 2πjf 1 sign πt −j (f) P+1 1 P+1 n δ (t − iT0) δ f − i=−∞ T0 n=−∞ T0 2 2 e−πt e−πf −at 1 e u(t); a > 0 a+2πjf −ajtj 2a e ; a > 0 a2+(2πf)2 Table 2: Some common Fourier transform pairs. 5 2 Signals 2.1 Continuous-time and discrete-time signals A continuous-time signal g(t) is a signal of the real variable t (time). For example, the signal cos(2πfct) is a function of time. A discrete-time signal g[n] is a sequence where the values of the index n are integers. Such a signal carries digital information. 2.2 Analog and digital signals An analog signal is a signal with a continuous range of amplitudes. A digital signal is a member of a set of M unique signals, dened over the T period, that represent M data symbols. Figure 2.1 represents an analog and a digital signal (top row). In practice, digital signals are materialized by a continuous-time signal, named a representation. Such representations are illustrated in Figure 2.1 (bottom row). Analog information signal Digital information signal 2 1.5 1 0.5 1 0 1 0 1 1 0 -0.5 -1 0 2 4 6 8 10 Analog representation Digital representation 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0 0.5 0 -0.5 -0.5 -1 0 -1 -1.5 -2 -0.5 -1.5 0 2 4 6 8 10 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Figure 2.1: Examples of representations of an continuous-time signal (left) and a discrete-time signal (right). 2.3 Energy and power The power in watts [W ] delivered by a voltage signal v(t) in volts to a resistive load R in ohms [Ω] is given by v2(t) P = v(t) i(t) = = R i2(t) (2.1) R 6 where i(t) is the current. In communications, the value of the resistive load is normalized to 1 [Ω]. Therefore, the instanta- neous normalized power is given by p(t) = v2(t) = i2(t) (2.2) The averaged normalized power is given by 1 Z T P = lim v2(t) dt (2.3) T !1 2T −T A signal v(t) is called a power signal if and only if its averaged power is non-zero and nite, that is 0 < P < 1. Example 6. Power of a sinusoid A cos (2πfct). The averaged normalized power delivered to a 1 [Ω] load is Z T 1 2 2 P = lim A cos (2πfct) dt (2.4) T !1 2T −T A2 Z T = lim (1 + cos(4πfct) dt (2.5) T !1 4T −T A2 A2 sin (4πf T ) sin (−4πf T ) = + lim c − c (2.6) 2 T !1 4T 4πfc 4πfc A2 A2 sin (4πf T ) A2 = + lim c = (2.7) 2 2 T !1 4πfcT 2 Therefore, A2 is the power of a sinusoid . The sinusoid is a power signal. 2 A cos (2πfct) The energy in Joules of a voltage signal v(t) in volts is given by Z +1 E = v2(t) dt (2.8) −∞ A signal v(t) is called an energy signal if and only if its energy is non-zero and nite, that is 0 < E < 1. 2.4 Deterministic vs stochastic tools As shown in Figure 2.2, the deterministic or stochastic nature of signals depends on the signal and the location in the communication chain.