EE1: Introduction to Signals and Communications Professor Kin K. Leung EEE and Computing Departments Imperial College kin.leung@imperial.ac.uk Lecture One Course Aims To introduce: 1. How signals can be represented and interpreted in time and frequency domains 2. Basic principles of communication systems 3. Methods for modulating and demodulating signals to carry information from an source to a destination 2
Recommended text book B.P Lathi and Z. Ding, Modern Digital and Analog Communication Systems , Oxford University Press ● Highly recommended ● Well balanced book ● It will be useful in the future ● Slides based on this book, most of the figures are taken from this book 3 Handouts ● Copies of the transparencies ● Problem sheets and solutions ● Everything is on the web http://www.commsp.ee.ic.ac.uk/~kkleung/Intro_Signals_Comm 4
Communications Input signal Input Transmitted Received Output Output message signal signal signal message Input Output Channel Receiver Transmitter transducer transducer Distortion and noise 5 Classifications of Signals ● Continuous-time and discrete-time signals ● Analog and digital signals ● Periodic and aperiodic signals ● Energy and power signals ● Deterministic and probabilistic signals 6
Continuous-time and discrete-time signals ● A signal that is specified for every value of time t is a continuous-time signal ● A signal that is specified only at discrete values of t is a discrete-time signals 7 Periodic and aperiodic signals ● A signal g ( t ) is said to be periodic if for some positive constant T 0 , g ( t ) = g ( t + T 0 ) for all t ● A signal is aperiodic if it is not periodic Same famous periodic signals: sin ω 0 t , cos ω 0 t , e j ω0 t , where ω 0 = 2π/ T 0 and T 0 is the period of the function (Recall that e j ω0 t = cos ω 0 t + j sin ω 0 t ) 8
Periodic Signal A periodic signal g ( t ) can be generated by periodic extension of any segment of g ( t ) of duration T 0 9 Energy and power signal First, define energy ● The signal energy E g of g ( t ) is defined (for a real signal) as E g g 2 ( t ) dt . ● In the case of a complex valued signal g ( t ), the energy is given by 2 E g *( ) ( ) t g t dt g t ( ) dt . g ● A signal g ( t ) is an energy signal if E g < ∞ 10
Power A necessary condition for the energy to be finite is that the signal amplitude goes to zero as time tends to infinity. In case of signals with infinite energy (e.g., periodic signals), a more meaningful measure is the signal power. 1 T 2 2 P lim g t ( ) dt g T T 2 T A signal is a power signal if 1 T 2 2 0 lim g t ( ) dt T T T 2 A signal cannot be an energy and a power signal at the same time 11 Energy signal example Signal Energy calculation 0 E g dt 4 e t dt 4 4 8. g 2 ( t ) (2) 2 dt 1 0 12
Power signal example Assume g ( t ) = A cos (ω 0 t + θ ) , its power is given by 1 T 2 2 2 P lim A cos ( w t ) dt g 0 T T 2 T 2 1 A T 2 lim 1 cos(2 w t 2 ) dt 0 T 2 T 2 T 2 2 A A T 2 T 2 lim dt lim cos(2 w t 2 ) dt 0 2 T 2 T T T 2 T T 2 2 A 2 13 Power of Periodic Signals Show that the power of a periodic signal g ( t ) with period T 0 is 1 T 2 2 0 P g t ( ) dt g T T 2 0 0 Another important parameter of a signal is the time average : 1 T 2 g average lim g ( t ) dt . T T 2 T 14
Deterministic and probabilistic signals ● A signal whose physical description is known completely is a deterministic signal. ● A signal known only in terms of probabilistic descriptions is a random signal. 15 Useful Signals: Unit impulse function The unit impulse function or Dirac function is defined as ( ) 0 t t 0 Area = 1 ( ) t d t 1 Δ→0 Multiplication of a function by an impulse: g t ( ) ( t T ) ( g T ) ( t T ) g t ( ) ( t T dt ) ( g T ). 16
Useful Signals: Unit step function Another useful signal is the unit step function u ( t ) , defined by 1 t 0 1 u t ( ) 0 t 0 Observe that t 1 t 0 t ( ) d = 0 t 0 Therefore du ( t ) ( t ). dt Use intuition to understand this relationship: The derivative of a ’unit step jump’ is an unit impulse function. 17 Useful Signals: Sinusoids Consider the sinusoid x ( t ) = C cos(2π f 0 t + θ ) f 0 (measured in Hertz) is the frequency of the sinusoid and T 0 = 1/ f 0 is the period. Sometimes we use ω 0 (radiant per second) to express 2π f 0 . Important identities 1 1 jx jx jx jx jx e cos x j sin , cos x x e e , sin x e e , 2 2 j 1 cos cos x y cos( x y ) cos( x y ) 2 a cos x b sin x C cos( x ) tan 1 b a 2 b 2 C with and a 18
Signals and Vectors ● Signals and vectors are closely related. For example, A vector has components - A signal has also its components - ● Begin with some basic vector concepts ● Apply those concepts to signals 19 Inner product in vector spaces x is a certain vector. It is specified by its magnitude or length | x | and direction. x Consider a second vector y . We define the inner or scalar product of two vectors as y θ < y , x > = | x || y | cos θ . Therefore, | x | 2 = < x , x >. When < y , x > = 0, we say that y and x are orthogonal (geometrically, θ = π/2 ). 20
Signals as vectors The same notion of inner product can be applied for signals. What is the useful part of this analogy? We can use some geometrical interpretation of vectors to understand signals! Consider two (energy) signals x ( t ) and y ( t ) . The inner product – correlation integral - is defined as x ( t ), y ( t ) x ( t ) y ( t ) dt For complex signals x ( t ), y ( t ) x ( t ) y *( t ) dt where y* ( t ) denotes the complex conjugate of y ( t ) . Two signals are orthogonal if < x ( t ), y ( t )> = 0. 21 Energy of orthogonal signals If vectors x and y are orthogonal, and if z = x + y z y | z | 2 = | x | 2 + | y | 2 (Pythagorean Theorem). x If signals x ( t ) and y ( t ) are orthogonal and if z ( t ) = x ( t ) + y ( t ) then E z = E x + E y . Proof for real x ( t ) and y ( t ) : 2 E ( ( ) x t y t ( )) dt z 2 2 x t dt ( ) y t dt ( ) 2 x t y t dt ( ) ( ) E E 2 x t y t dt ( ) ( ) x y E E x y dt 0. since x ( t ) y ( t ) 22
Power of orthogonal signals The same concepts of orthogonality and inner product extend to power signals. For example, g ( t ) = x ( t ) + y ( t ) = C 1 cos( ω 1 t + θ 1 ) + C 2 cos( ω 2 t + θ 2 ) and ω 1 ≠ ω 2 . 2 2 x C 1 y C 2 P 2 , P 2 . The signal x ( t ) and y ( t ) are orthogonal: < x ( t ) , y ( t ) > = 0. Therefore, 2 2 y C 1 2 C 2 g P x P P 2 . 23 Signal comparison: Correlation If vectors x and y are given, we have the correlation measure as y , x c n cos x y Clearly, −1 ≤ c n ≤ 1 . In the case of energy signals: 1 c y t x t dt ( ) ( ) n E E y x again −1 ≤ c n ≤ 1 . 24
Best friends, worst enemies and complete strangers ● c n = 1 . Best friends . This happens when g ( t ) = Kx ( t ) and K is positive. The signals are aligned, maximum similarity. ● c n = −1 . Worst Enemies . This happens when g ( t ) = Kx ( t ) and K is negative. The signals are again aligned, but in opposite directions. The signals understand each others, but they do not like each others. ● c n = 0 . Complete Strangers The two signals are orthogonal. We may view orthogonal signals as unrelated signals. 25 Correlation Why do we bother poor undergraduate students with correlation? Correlation is widely used in engineering. For instance ● To design receivers in many communication systems ● To identify signals in radar systems ● For classifications 26
Correlation examples Find the correlation coefficients between: ● x ( t ) = A 0 cos( ω 0 t ) and y ( t ) = A 1 sin( ω 1 t ) c x,y = 0. ● x ( t ) = A 0 cos( ω 0 t ) and y ( t ) = A 1 cos( ω 1 t ) and ω 0 ≠ ω 1 c x,y = 0. ● x ( t ) = A 0 cos( ω 0 t ) and y ( t ) = A 1 cos( ω 0 t ) c x,y = 1. ● x ( t ) = A 0 sin( ω 0 t ) and y ( t ) = A 1 sin( ω 1 t ) and ω 0 ≠ ω 1 c x,y = 0. ● x ( t ) = A 0 sin( ω 0 t ) and y ( t ) = A 1 sin( ω 0 t ) c x,y = 1. ● x ( t ) = A 0 sin( ω 0 t ) and y ( t ) = −A 1 sin( ω 0 t ) c x,y = -1. 27 Signal representation by orthogonal signal sets ● Examine a way of representing a signal as a sum of orthogonal signals ● We know that a vector can be represented as the sum of orthogonal vectors ● The results for signals are parallel to those for vectors ● Review the case of vectors and extend to signals 28
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