. Introduction to Signals and Operations Model of a Communication System [] Figure. (a) Model of a communication system, and (b) signal processing functions. Classification of Signals. Continuous-time and discrete-time signals [2] By the term continuous-time signal we mean a real or complex function of time s(t), where the independent variable t is continuous. If t is a discrete variable, i.e., s(t) is defined at discrete times, then the signal s(t) is a discrete-time signal. A discrete-time signal is often identified as a sequence of numbers, denoted by {s(n)}, where n is an integer. 2. Analogue and digital signals [2] If a continuous-time signal s(t) can take on any values in a continuous time interval, then s(t) is called an analogue signal. If a discrete-time signal can take on only a finite number of distinct values, {s(n)}, then the signal is called a digital signal. 3. Deterministic and random signals [2] Deterministic signals are those signals whose values are completely specified for any given time. Random signals are those signals that take random values at any given times. 4. Periodic and nonperiodic signals [2] A signal s(t) is a periodic signal if s(t) = s(t + n 0 ), where 0 is called the period and the integer n > 0. If s(t) s(t + 0 ) for all t and any 0, then s(t) is a nonperiodic or aperiodic signal..
5. Power and energy signals [2, 3] A complex signal s(t) is a power signal if the average normalised power P is finite, where /2 0 < P = lim s(t)s*(t)dt < (.) /2 and s*(t) is the complex conjugate of s(t). A complex signal s(t) is an energy signal if the normalised energy E is finite, where 0 < E = s(t)s*(t) dt = s(t) 2 dt < (.2) In communication systems, the received waveform is usually categorised into the desired part, containing the information signal, and the undesired part, called noise. Some Useful Functions. Unit impulse function [2] he unit impulse function, also known as the dirac delta function, δ(t), is defined by s(t) δ(t) dt = s(0) (.3) An alternative definition [4] is δ(t) dt = (.4a) and δ(t) =, t = 0 0, t 0 (.4b).2
Figure.2 Unit impulse function. 2. Unit step function [2] he unit step function u(t) is u(t) =, t > 0 0, t < 0 (.5) Figure.3 Unit step function. and the unit step function is related to the unit impulse function by u(t) = t δ(τ) dτ (.6) and du (t ) dt = δ(t) (.7) 3. Sampling function [4] A sampling function is denoted by Sa(x) = sin x x (.8) Figure.4 Sampling function. 4. Sinc function [4] A sinc function is denoted by sinc x = sin π x π x (.9) Hence, Sa(x) = sinc x ( π ) (.0).3
5. Rectangular function [4] A single rectangular pulse is denoted by Π( t )=, t 2 0, t > 2 (.) 6. riangular function [4] A triangular function is denoted by Λ( t )= t, t 0, t > (.2) Some Useful Operations [4]. ime average he time average operator is given by /2 <[.]> = lim [.]dt (.3) /2 If a waveform is periodic, the time average operator can be reduced to <[.]> = a + /2 0 [.]dt (.4) 0a /2 0 where 0 is the period of the waveform and a is an arbitrary real constant, which may be taken to be zero. Equation (.4) readily follows from (.3) because, referring to (.3), integrals over successive time intervals 0 seconds wide have identical area if the waveform is periodic. As these integrals are summed, the total area and are proportionally larger, resulting in a value for the time average that is the same as just integrating over one period and dividing by 0. In summary, (.3) may be used to evaluate the time average of any type of waveform..4
Equation (.4) is valid only for periodic waveforms. 2. Direct-current value he direct-current (dc) value of a waveform is given by /2 <s(t)> = lim s(t) dt (.5) /2 We can see that this is the time average of s(t). Over a finite interval of interest, the dc value is t 2 <s(t)> = t t s(t) dt (.6) 2 t 3. Power and energy he instantaneous power (incremental work divided by incremental time) is given by p(t) = v(t) i(t) (.7) where v(t) denotes voltage and i(t) denotes current. he average power is given by P = <p(t)> = <v(t) i(t)> (.8) he root mean square (rms) value of s(t) is given by S rms = < s 2 (t ) > (.9) If a load is resistive, the average power is given by P = <v 2 (t ) > R = <i 2 (t)>r = V 2 rms R 2 = I rms R = Vrms I rms (.20) where R is the value of the resistive load. When R = Ω, P becomes the normalised.5
power. he average normalised power of a real-valued signal s(t) is given by P = <s 2 /2 (t)> = lim s 2 (t)dt (.2) /2 he total normalised energy of a real-valued signal s(t) is given by /2 E = lim s 2 (t)dt (.22) /2 4. Decibel he decibel gain of a circuit is given by db = 0 log 0 P out P in (.23) If resistive loads are involved, (.23) can be reduced to db = 20 log V rms out 0 V rms in + 0 log R in 0 R load (.24) or db = 20 log 0 I rms out I rms in + 0 log 0 R load R in (.25) If normalised powers are used, db = 20 log V rms out 0 V rms in = 20 log I rms out 0 I rms in (.26) he decibel power level with respect to mw is given by.6
dbm = 0 log 0 actual power level in watts 0 3 (.27) he decibel power level with respect to W is given by dbw = 0 log 0 (actual power level in watts) he decibel power level with respect to a mv rms level is given by dbmv = 20 log V rms 0 0 3 (.28) 5. Complex number and phasor A complex number c is said to be a phasor if it is used to represent a sinusoidal waveform. hat is, s(t) = c cos [ω 0 t + c] = Re {c e jω 0 t } (.29) where the phasor c = c e jφ (.30) and φ = c. Other Useful Operations. Cross-correlation [5] he cross-correlation of two real-valued power waveforms s (t) and s 2 (t) is defined by /2 R (τ) = <s (t) s 2 (t+ τ)> = lim 2 s (t) s 2 (t+ τ) dt /2 (.3) If s (t) and s 2 (t) are periodic with the same period 0, then.7
R 2 (τ) = /2 0 s (t) s 2 (t+ τ) dt (.32) 0 /2 0 he cross-correlation of two real-valued energy waveforms s (t) and s 2 (t) is defined by R 2 (τ) = s (t) s 2 (t+ τ) dt (.33) Correlation is a useful operation to measure the similarity between two waveforms. o compute the correlation between waveforms, it is necessary to specify which waveform is being shifted. In general, R (τ) is not equal to R (τ), where 2 2 R 2 (τ) = <s 2 (t) s (t+ τ)>. he cross-correlation of two complex waveforms is R 2 (τ) = <s *(t) s 2 (t+ τ)>. 2. Auto-correlation [2, 4] he auto-correlation of a real-valued power waveform s (t) is defined by /2 R (τ) = <s (t) s (t+ τ)> = lim s (t) s (t+ τ) dt /2 (.34) If s (t) is periodic with fundamental period 0, then R (τ) = /2 0 s (t) s (t+ τ) dt (.35) 0 /2 0 he auto-correlation of a real-valued energy waveform s (t) is defined by R (τ) = s (t) s (t+ τ) dt (.36).8
he auto-correlation of a complex power waveform is R (τ) = <s *(t)s (t+ τ)>. 3. Convolution [4] he convolution of a waveform s (t) with a waveform s 2 (t) is given by s 3 (t) = s (t) * s 2 (t) = s (λ) s 2 (t-λ) dλ s 3 (t) = s (t) * s 2 (t) = s (λ) s 2 [-(λ-t)] dλ (.37a) (.37b) where * denotes the convolution operation. (.37b) is obtained by. ime reversal of s 2 (t) to obtain s 2 (-λ). 2. ime shifting of s 2 (-λ) to obtain s 2 [-(λ-t)]. 3. Multiplying s (λ ) and s 2 [-(λ-t)] to form the integrand s (λ) s 2 [-(λ-t)]. Example. Convolution of a rectangular waveform s (t) =, 0 < t < 0, elsewhere with an exponential waveform s 2 (t) = e -t/ u(t). Figure.5 Convolution between a rectangular waveform and an exponential waveform. References [] B. Sklar, Digital Communications, Prentice Hall, 998. [2] H. P. Hsu, Analog and Digital Communications, McGraw-Hill, 993. [3] J. D. Gibson, Modern Digital and Analog Communications, 2/e, Macmillan Publishing Company, 993. [4] L. W. Couch II, Digital and Analog Communication Systems, 5/e, Prentice Hall, 997. [5] H. aub, and D. L. Schilling, Principles of Communication Systems, 2/e, McGraw-Hill, 986..9
From other sources Source ADC Source encoder Cipher Channel encoder Multiplex Modulator Frequency spreader Multiple access... ransmitter Analogue signal Digital signals Synchroniser Channel... Receiver Sink DAC Source decoder Demodulator Decipher Channel decoder Demultiplex Frequency despreader Multiple access (a) o other destinations Essential ADC/ Source coding PCM DPCM DM Synthesis coding ransform coding Encryption Block cipher Stream cipher Channel coding Block Convolutional Synchronisation Carrier Subcarrier Symbol Frame Bit Multiplex/ Multiple access FDM/FDMA DM/DMA CDM/CDMA (b) Modem Coherent PSK Coherent FSK Coherent ASK CPM DPSK Noncoherent FSK Noncoherent ASK CPM Frequency spreading Direct sequence Frequency/ ime hopping Figure. (a) Model of a communication system, and (b) signal processing functions..0
δ ( t ) t 0 Figure.2 Unit impulse function. u ( t ) t 0 Figure.3 Unit step function. sin x x - 2π - π 0 π 2π Figure.4 Sampling function. x.
s ( λ ) - 0 2 s 2 ( λ ) - 0 2 s 2 (-( λ - t )) - 0 t 2 λ λ λ s ( t ) 3 0.63-0 2 t Figure.5 Convolution of a rectangular waveform and an exponential waveform..2