Signals and Systems: Introduction Prof. Miguel A. Muriel Siggynals and Systems: Introduction 1- Introduction to Signals and Systems 2- Continuous-Time Signals 3- Discrete-Time Signals 4- Description of Systems 5- Time-Domain System Analysis 6- Spectral Method 7- Fourier Transform 8- Sampling 9- Discrete Fourier Transform (DFT) Miguel A. Muriel - Signals and Systems: Introduction - 2 1- Introduction to Siggynals and Systems Miguel A. Muriel - Signals and Systems: Introduction - 3 Introduction -A signal is any physical phenomenon which conveys ifinformati on. -Systems respondilddd to signals and produce new si ilgnals. -Excitation signals are applied at system inputs and response signals are produced at system outputs. Miguel A. Muriel - Signals and Systems: Introduction - 4 Siggypnal types Continuous-Time Discrete-Time x(t) Continuous-Value x[n] Continuous-Value Signal Signal t n Continuous-Time Continuous-Time x(t) Discrete-Value Discrete-Value Signal x(t) Signal t t Digital Signal Continuous-Time CtiContinuous-VlValue x(t) Random Signal x(t) Noisy Digital Signal t t Miguel A. Muriel - Signals and Systems: Introduction - 5 Conversions Between Signal Types Miguel A. Muriel - Signals and Systems: Introduction - 6 Syypstem example -A communication system has an information signal plus noise signals. Miguel A. Muriel - Signals and Systems: Introduction - 7 Vo ice Signal Miguel A. Muriel - Signals and Systems: Introduction - 8 (a) 32 ms (b) 256 samples (a) Segment of a continuous-time speech signal x(t). (b) Sequence of samples x[][n]=x(nT) obta ined from the silignal in part ()(a) with T =125 μs. Miguel A. Muriel - Signals and Systems: Introduction - 9 2- Continuous-Time Signals Miguel A. Muriel - Signals and Systems: Introduction - 10 -All continuous signals that are functions of time are continuous-time , but not all continuous-time signals are continuous functions of time. Miguel A. Muriel - Signals and Systems: Introduction - 11 Impulse ()t tt 000, 0 1 tdt 1 0 t 2 2 -The imppyulse is not a function in the ordinary sense because its value at the time of its occurrence is not defined. -It is a functional. Miguel A. Muriel - Signals and Systems: Introduction - 12 -It is representdted graphi hillcally btilby a vertical arrow. -Its strength is either written beside it or is represented by its length. Miguel A. Muriel - Signals and Systems: Introduction - 13 Proppperties of the Impulse Sampling Property ft tdtf 0 ft t t dtft 00 -The samppgppyling property “extracts” the value of a function at a point. Miguel A. Muriel - Signals and Systems: Introduction - 14 Scaling Property 1 at t00 t t a -This property illustrates that the impulse is different from ordinary mathematical functions . Miguel A. Muriel - Signals and Systems: Introduction - 15 Periodic Impulse T ttnTn an integer n -The periodic impulse is a sum of infinitely many uniformly- spaced impulses . Miguel A. Muriel - Signals and Systems: Introduction - 16 Miguel A. Muriel - Signals and Systems: Introduction - 17 Step 101 , t 0 u1/2tt , 0 000 , t 0 Precise Graph Commonly-Used Graph Miguel A. Muriel - Signals and Systems: Introduction - 18 Signum 1 , t 0 sgntt 0 , 0 101 , t 0 Precise Graph Commonly-Used Graph Miguel A. Muriel - Signals and Systems: Introduction - 19 Ramp tt,0 t ramptudtut 0,t 0 Miguel A. Muriel - Signals and Systems: Introduction - 20 Pulse ((g)Rectangle) 1 ,t 1/2 rectt 1/2 , t1/2 ut 1/2 ut 1/2 01/20 , t 1/2 Miguel A. Muriel - Signals and Systems: Introduction - 21 Real Sinusoid and Real Exponential Miguel A. Muriel - Signals and Systems: Introduction - 22 Comppplex Exponential t gt( ) Ae e j2 ft e jft2 cos 2ft j sin 2 ft e jft2 cos 2ft j sin 2 ft eejft22 jft cos 2 ft 2 eejft22 jft sin 2 ft 2 j Miguel A. Muriel - Signals and Systems: Introduction - 23 Miguel A. Muriel - Signals and Systems: Introduction - 24 Scaling and Shifting tt 0 gt Ag a Miguel A. Muriel - Signals and Systems: Introduction - 25 gt Agbtt0 Miguel A. Muriel - Signals and Systems: Introduction - 26 Differentiation dx() t gt dt Miguel A. Muriel - Signals and Systems: Introduction - 27 Integration t gt() x d Miguel A. Muriel - Signals and Systems: Introduction - 28 Even and Odd Signals Even Functions Odd Functions gtgt gt gt Miguel A. Muriel - Signals and Systems: Introduction - 29 Even and Odd Parts of Signals gt geo t g t gt g t The even part of a function is gt e 2 gt g t The odd part of a function is gt o 2 -The derivative of an even/odd function is odd/even. -The integral of an even/odd function is an odd/even function, plus a constant. Miguel A. Muriel - Signals and Systems: Introduction - 30 Periodic Signals gggt g tnT n is any integer T ifthftiis a peridiod of the function -The minimum positive value of TgtgtT for which is called the fundamental period T0 of the function. -The reciprocal of the fundamental period is the fundamental frequency fT001/ Miguel A. Muriel - Signals and Systems: Introduction - 31 Examples of periodic functions with fundamental period T0 Miguel A. Muriel - Signals and Systems: Introduction - 32 Sigggynal Energy The signal energy of a signal xt is 2 Extdt x Miguel A. Muriel - Signals and Systems: Introduction - 33 Signal Power -Some signals have infinite signal energy. -This usually occurs because the signal is not time limited. -In that case it is more convenient to deal with average signal power. -The averagggpe signal power of a signal xt is: T /2 1 2 Pxtdtx lim T T T /2 Miguel A. Muriel - Signals and Systems: Introduction - 34 -For a periodic signal xtt the average sign al power is: 1 2 Pxtdt x T T 2 where Txt is any period of Miguel A. Muriel - Signals and Systems: Introduction - 35 -A silihfiiilignal with finite signal energy i s called an energy signal. -A signal with infinite signal energy and finite average signal power is called a power signal. -AllidiilAll periodic signals are power si ilgnals. Miguel A. Muriel - Signals and Systems: Introduction - 36 3- Discrete-Time Signals Miguel A. Muriel - Signals and Systems: Introduction - 37 Sampling and Discrete Time -Sampling is the acquisition of the values of a continuous-time signal at discrete po ints i n ti me. -xt is a continuous-time signal, xn is a discrete-time signal. xn xnT s Ts is the t ime between samples (a) An ideal sampler (b) An ideal sampler sampling uniformly Miguel A. Muriel - Signals and Systems: Introduction - 38 Creating a discrete-time signal by sampling a continuous-time signal Miguel A. Muriel - Signals and Systems: Introduction - 39 Unit Impulse 1,n 0 n 0,n 0 -The discrete-time unit impulse is a function in the ordinary sense (in contrast with the continuous-time impulse). -It has a sampling property AnnxnAxn 00 n -But no scaling property, nan for any non-zero, finite integer a. Miguel A. Muriel - Signals and Systems: Introduction - 40 Periodic Impulse N nnmN m Miguel A. Muriel - Signals and Systems: Introduction - 41 Unit Sequence 1,n 0 un 0,n 0 Miguel A. Muriel - Signals and Systems: Introduction - 42 Signum 1 , n 0 sgnnn 0 , 0 101 , n 0 Miguel A. Muriel - Signals and Systems: Introduction - 43 Unit Ramp nn, 0 ramp n nu n 0,n 0 Miguel A. Muriel - Signals and Systems: Introduction - 44 Exponentials n xn A A is a real constant Real Complex Miguel A. Muriel - Signals and Systems: Introduction - 45 Sinusoids gn Acos 2 Fn0 A is a real constant, is a real phas e shift in radians Fn0 is a real number, and is discrete time -Unlike a continuous-time sinusoid, a dittiiiscrete-time sinusoid idit is not necessaril yperidiiodic. -gnF periodic 0 must be a ratio of integers (a rational number). Miguel A. Muriel - Signals and Systems: Introduction - 46 Four discrete-time sinusoids Miguel A. Muriel - Signals and Systems: Introduction - 47 -Two sinusoids whose analytical expressions look different gn101202 Acos 2 Fn and gn Acos 2 Fn may actually be the same if FFm02 01 , where m is an integer. AFnAFnmnAFcos 202 cos 2 01 2 cos 2 01n Miguel A. Muriel - Signals and Systems: Introduction - 48 Two cosines with different F’s but the same functional behavior Miguel A. Muriel - Signals and Systems: Introduction - 49 A discrete-time sinusoid with frequency F repeats every time F changes by one Miguel A. Muriel - Signals and Systems: Introduction - 50 Scaling and Shifting Functions Let gn be graphically defined by: gn 0150 , n 15 Miguel A. Muriel - Signals and Systems: Introduction - 51 Time shifting nnnn00 an integer Miguel A. Muriel - Signals and Systems: Introduction - 52 Time compression nKnK an integer > 1 Time expansion nnKK / an integer > 1 -For all nnKgnK such that / is an integer, / is defined. -Fllhh/iFor all nn such that / KKK is not an i nteger, gn / K idfidis not defined. Miguel A. Muriel - Signals and Systems: Introduction - 53 Time compression for a discrete-time function Miguel A. Muriel - Signals and Systems: Introduction - 54 Time expansion for a discrete-time function (K=2) Miguel A. Muriel - Signals and Systems: Introduction - 55 Differencing gn xn xn 1 gn xn 1 xn Miguel A. Muriel - Signals and Systems: Introduction - 56 Accumulation n gn hm m Miguel A. Muriel - Signals and Systems: Introduction - 57 Even and Odd Signals gn gn gn g n gn gn gn g n gn gn eo22 Miguel A. Muriel - Signals and Systems: Introduction - 58 Periodic Signals gn gn mN m is any integer N ifthftiis a peridiod of the function -The minimum positive value of NgngnN for which is called the fundamental period N0 of the function. -The reciprocal of the fundamental period is the fundamental frequency FN001/ Miguel A.