Outline. Plotting discrete-time signals. Sampling Process. Discrete-Time Signal Representations Important D-T Signals Digital Signals

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1 Outlie Discrete-Time Sigals-Itroductio. Plottig discrete-time sigals. Samplig Process. Discrete-Time Sigal Represetatios Importat D-T Sigals Digital Sigals Discrete-Time Sigals-Itroductio The time variable t is said to be a discrete-time variable if t takes o oly the discrete values t t for some rage of iteger values of. For example: t t for 0,,,..., Discrete-time sigal: is a sigal that is a fuctio of the discrete-time variable t ; i other words, a discrete-time sigal has defied values oly at the discretetime poits t t ; so, a discrete time sigal is a sequece of umbers idexed by itegers. Example: x [..., 3,,,0,,,3,..., brackets idicates D-T sigal, parethesis idicates C-T sigal. Plottig discrete-time sigals A stem plot emphasizes that the sigal does ot exist i-betwee iteger values. Sometimes we plot with lie segmets coectig the dots. Matlab example: x [ is give by: x [ 3], x[ ], x[ ] 3, x[0] 5, x[], x[], x[3] 7 with x [ 0 for all other. A plot of this sigal (see Figure _) ca be geerated by the followig Matlab commads (see Chapter_.m): % script Chapter _.m % Plot a discrete-time sigal usig Matlab % we eed two vectors to plot oe-dimesioal sigal % the first vector defies the horizotal axes: % samples poits to calculate the sigal values. % the secod oe defies the values of the sigal % at samples poits (vertical axes) = -3:3; %first vector %x[=0 for all other. x = [,-,-3,5,,-,7]; %secod vector stem(,x,'filled'); xlabel('time Samples: ');

ylabel('x[: Sigal values'); title('discrete-time Sigal'); axis([-4 4-4 8]); Figure _ Samplig Process oe of the most commo ways i which D-T sigal arise is i samplig the C-T sigals.

2 ylabel('x[: Sigal values'); title('discrete-time Sigal'); axis([ ]); Figure _ Samplig Process oe of the most commo ways i which D-T sigal arise is i samplig the C-T sigals. We ca describe the samplig process as a switch that closes briefly every T secods (as show i figure _), the output of the switch ca be viewed as a D-T sigal that is a fuctio of the discrete time poits t t, where...,,,0,,,..., C-T Sigal D-T Sigal Switch Figure _: Samplig Process The resultig D-T sigal is called the sampled versio of the origial C-T sigal, ad T is called the samplig iterval. Samplig methods: Uiform Samplig (T - costa Nouiform Samplig (T - variable) By defiitio of the samplig process, the value of x [ for ay iteger value of is give by x[ x( t T x( T) is called samplig frequecy or samplig rate ( F s ) i samples/secods. T Importat Questio: How fast should we sample a specific sigal? We should sample a specific sigal with samplig rate that is slightly more tha twice the highest frequecy i this sigal. Example: CD audio is sampled at 4400 samples per secod T.69 sec, because the humas ca't hear frequecies above 4400 approximately 0 khz.

3 Discrete-Time Sigal Represetatios Graphical Represetatio: as show i figures _. Fuctioal Represetatio, such as,4 x[ 3,3 0 otherwise Tabular Represetatio, such as x [ Sequece Represetatio, such as A ifiite-duratio sigal or sequece with the time origi ( 0 ) idicated by the symbol is represeted as: x [...,0,0,0,3, 6,3,,,5,... A sequece x [ which is zero for 0, ca be represeted as x [ 0,,4,0,0,3,... Matlab example : % script Chapter _.m % Plot a discrete-time sigal % x[=3*exp(-0.3)si(/3)(-3)^ = 0:0; %x[=0 for all other. x = 3*exp(-0.3*).*si(/3*).*(-3).^; stem(,x,'filled'); xlabel(''); ylabel('x['); title('d-t Sigal: x[=3exp(-0.3)si(/3)(-3)^'); axis auto; Figure _3 3

4 Importat D-T Sigals Much of what we leared about C-T sigals carries over to D-T sigals.. D-T Uit Step sigal D-T uit step sigal u [ which is defied by, 0,,,3,... u [ 0,,... D-T step sigal ca be obtaied by samplig the C-T step u ( (sampled versio of u (), the sketch of this sigal is show i figure _4 ad the Matlab code to geerate uit step sigal is writte i Chapter _3.m %Script Chapter_3.m fuctio uitstep(p) % Geerates ad plots x[ = u[; % % UNITSTEP (NP) %p poits' cout if p < 0 error('argumet p must satisfy p > 0') ed = [0:p]; x = [oes(,p+)]; stem(,x,'filled'); xlabel(''); ylabel('x['); title('d-t Uit Step Sigal') axis ([- p+ 0 ]); grid; Figure _4. D-T Uit Ramp sigal D-T uit ramp sigal r [ which is defied by 0,,,... r [ 0,,... See figure _5a 4

5 If the uit ramp r( t u( is sampled, the result is give by r[ r( t T r( T) See figure _5b These two sigals i figure _5 are ot the same. Uless the samplig iterval T is equal to. r[ 4 3 r[ 4T 3T T T Figure _5a Figure _5b Figure _5: D-T Uit-Ramp Sigal 3. Uit Pulse (D-T Impulse ) There is o sampled versio of the uit impulse ( sice (0) is ot defied. The uit-impulse sigal, defied by, 0 [ 0, 0 See figure _6. [ 3 4 Figure _6: D-T Uit-Impulse Sigal Siftig property for D-T Delta fuctio Note: [ works iside summatio, the same way ( works iside itegral Compare [ ( dt Compare x[ [ 0 ] x[ 0 ] x( ( t t0 ) dt x( t0 ) Ay sequece ca be expressed as: k 5 x [ x[ k] [ k] 4. Periodic D-T sigals (D-T Siusoid) A discrete sigal x [ is periodic if there exists a positive iteger r such that x[ r] x[, for all iteger.

6 D-T siusoid give by: x[ Acos -use upper case omega for frequecy of D-T sigals. D-T frequecy i radias per uit time T what is the uit for? must be i radias. is "how may radias jump for each sample ", is i radias /sample is the phase i radias. The sigal is periodic with period r if Acos ( r) Acos Recall that cosie fuctio repeats every radias, so that Acos Acos q for all iteger q, so q r q (Fudametal period). r 5. D-T Rectagular Pulse Let S be a positive odd iteger. D-T rectagular pulse sigal P S [ of legth S defied by ( S ) ( S ),...,,0,,..., P S [ 0 all other A graphical represetatio of this sigal is illustrated i figure -7. P S [ ( ) S ( S ) Figure _7: D-T Rectagular Sigal Digital Sigals A digital sigal x [ is a discrete-time sigal whose values belog to the fiite set: a, a,..., a N, at each time istat t, we have x( t ) x[, for some j, where j N. a j A practical ADC ot oly gives a D-T sigal but also oe that is "Digital". Biary sigal is a digital sigal whose values are equal to or 0: x [ 0 or, for...,,,0,,,... The sampled uit-step fuctio ad uit-pulse fuctio are examples of biary sigals. 6

Limits of sequences. Contents 1. Introduction 2 2. Some notation for sequences The behaviour of infinite sequences 3

Limits of sequences. Contents 1. Introduction 2 2. Some notation for sequences The behaviour of infinite sequences 3 Limits of sequeces I this uit, we recall what is meat by a simple sequece, ad itroduce ifiite sequeces. We explai what it meas for two sequeces to be the same, ad what is meat by the -th term of a sequece.