Discrete-Time Signals and Systems. Introduction to Digital Signal Processing. Independent Variable. What is a Signal? What is a System?

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1 Discree-Time Sigals ad Sysems Iroducio o Digial Sigal Processig Professor Deepa Kudur Uiversiy of Toroo Referece: Secios. -.4 of Joh G. Proakis ad Dimiris G. Maolakis, Digial Sigal Processig: Priciples, Algorihms, ad Applicaios, 4h ediio, 7. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5. Sigals, Sysems ad Sigal Processig Wha is a Sigal? Wha is a Sysem? Idepede Variable. Sigals, Sysems ad Sigal Processig Sigal: ay physical quaiy ha varies wih ime, space, or ay oher idepede variable or variables Examples: pressure as a fucio of aliude, soud as a fucio of ime, color as a fucio of space,... x() = cos(π), x() = 4 + 3, x(m, ) = (m + ) Sysem: a physical device ha performs a operaio o a Examples: aalog amplifier, oise caceler, commuicaio chael, rasisor,... y() = 4x(), dy() d + 3y() = dx() d + x(), y() y( ) = 3x() + x( ) A ca be represeed as a fucio x() ad cosiss of:. oe or more depede variable compoes (e.g., air pressure x, R-G-B color [x x x 3 ] T );. oe or more idepede variables (e.g., ime, 3-D spacial locaio (s, s, s 3 )). Please oe: i his course we will ypically use ime o represe he idepede variable alhough i geeral i ca correspod o ay oher ype of idepede variable. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 3 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 4 / 5

2 . Classificaio of Sigals Coiuous-Time versus Discree-Time Sigals. Classificaio of Sigals Coiuous-Time versus Discree-Time Sigals Coiuous-Time Sigals: is defied for every value of ime i a give ierval (a, b) where a ad b. Examples: volage as a fucio of ime, heigh as a fucio of pressure, umber of posiro emissios as a fucio of ime. Discree-Time Sigals: is defied oly for cerai specific values of ime; ypically ake o be equally spaced pois i a x() x() ierval. Examples: umber of socks raded per day, average icome per provice x() x() x[] x[] x[] x[] Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 5 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5. Classificaio of Sigals Coiuous-Ampliude versus Discree-Ampliude. Classificaio of Sigals Coiuous-Ampliude versus Discree-Ampliude Coiuous-Ampliude Sigals: ampliude akes o a specrum of values wihi oe or more iervals Examples: color, emperaure, pai-level Discree-Ampliude Sigals: ampliude akes o values from a fiie se Examples: digial image, populaio of a coury - x() x[] x() x[] x() x[] x() x[] Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 7 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 8 / 5

3 Aalog ad Digial Sigals. Classificaio of Sigals Aalog ad Digial Sigals. Classificaio of Sigals aalog = coiuous-ime + coiuous ampliude digial = discree-ime + discree ampliude coiuous ampliude discree ampliude Aalog s are fudameally sigifica because we mus ierface wih he real world which is aalog by aure. coiuous-ime x() x() discree-ime x[] x[] Digial s are impora because hey faciliae he use of digial processig (DSP) sysems, which have pracical ad performace advaages for several applicaios. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 9 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5 Aalog ad Digial Sysems. Classificaio of Sigals. Classificaio of Sigals Deermiisic vs. Radom Sigals aalog sysem = aalog ipu + aalog oupu advaages: easy o ierface o real world, do o eed A/D or D/A coverers, speed o depede o clock rae Deermiisic : ay ha ca be uiquely described by a explici mahemaical expressio, a able of daa, or a well-defied rule pas, prese ad fuure values of he are kow precisely wihou ay uceraiy digial sysem = digial ipu + digial oupu advaages: re-cofigurabiliy usig sofware, greaer corol over accuracy/resoluio, predicable ad reproducible behavior Radom : ay ha lacks a uique ad explici mahemaical expressio ad hus evolves i ime i a upredicable maer i may o be possible o accuraely describe he he deermiisic model of he may be oo complicaed o be of use. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5

4 Wha is a pure frequecy? Wha is a pure frequecy? x a () = A cos(ω + θ) = A cos(πf + θ), R aalog, A x a () A ad < < A = ampliude Ω = frequecy i rad/s F = frequecy i Hz (or cycles/s); oe: Ω = πf θ = phase i rad Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 3 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 4 / 5 Coiuous-ime Siusoids Coiuous-ime Siusoids: Frequecy x a () = A cos(ω + θ) = A cos(πf + θ), R. for F R, x a () is periodic i.e., here exiss Tp R + such ha x a () = x a ( + T p ). disic frequecies resul i disic siusoids i.e., for F F, A cos(πf + θ) A cos(πf + θ) 3. icreasig frequecy resuls i a icrease i he rae of oscillaio of he siusoid i.e., for F < F, A cos(πf + θ) has a lower rae of oscillaio ha A cos(πf + θ) smaller F, larger T larger F, smaller T Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 5 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5

5 Discree-ime Siusoids Discree-ime Siusoids x() = A cos(ω + θ) = A cos(πf + θ), Z x() = A cos(ω + θ) = A cos(πf + θ), Z discree-ime (o digial), A x a () A ad Z A = ampliude ω = frequecy i rad/sample f = frequecy i cycles/sample; oe: ω = πf θ = phase i rad. x() is periodic oly if is frequecy f is a raioal umber Noe: raioal umber is of he form k k for k, k Z periodic discree-ime siusoids: x() = cos( 4 7 π), x() = si( π 5 + 3) aperiodic discree-ime siusoids: x() = cos( 4 7 ), x() = si( π + 3) Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 7 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 8 / 5 Discree-ime Siusoids x() = A cos(ω + θ) = A cos(πf + θ), Z MINIMUM OSCILLATION. radia frequecies separaed by a ieger muliple of π are ideical or cyclic frequecies separaed by a ieger muliple are ideical 3. lowes rae of oscillaio is achieved for ω = kπ ad highes rae of oscillaio is achieved for ω = (k + )π, for k Z subsequely, his correspods o lowes rae for f = k (ieger) ad highes rae for f = k+ (half ieger), for k Z. MAXIMUM OSCILLATION MINIMUM OSCILLATION Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 9 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5

6 Complex Expoeials Complex Expoeials e jφ = cos(φ) + j si(φ) Euler s relaio cos(φ) = ejφ +e jφ si(φ) = ejφ e jφ j Coiuous-ime: Discree-ime: A e j(ω+θ) = A e j(πf+θ) A e j(ω+θ) = A e j(πf+θ) where j Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5 Periodiciy: Coiuous-ime Periodiciy: Discree-ime x() = x( + T ), T R + A e j(πf+θ) j(πf (+T )+θ) = A e e jπf e jθ = e jπf e jπft e jθ = e jπft e jπk = = e jπft, k Z T = k F k Z T = F, k = sg(f ) x() = x( + N), N Z + A e j(πf+θ) j(πf (+N)+θ) = A e e jπf e jθ = e jπf e jπfn e jθ = e jπfn e jπk = = e jπfn, k Z f = k N k Z N = k f, mi k Z such ha k f Z + Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 3 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 4 / 5

7 Example : ω = π/ = π x[] = cos ( π ) N = πk Ω = πk π = k N = for k = The fudameal period is which correspods o k = evelope cycles. ENVELOPE CYCLES Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 5 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig / 5 Example : ω = 8π/3 = π 8 3 x[] = cos N = πk Ω ( ) 8π 3 = πk π 8 3 N = 3 for k = 4 = 3 4 k The fudameal period is 3 which correspods o k = 4 evelope cycles. ENVELOPE CYCLES Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 7 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 8 / 5

8 Example 3: ω = / = π π ( x[] = cos ) N = πk Ω = πk = πk N Z + does o exis for ay k Z; x[] is o-periodic. NOT PERIODIC Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 9 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 3 / 5 Uiqueess: Coiuous-ime Uiqueess: Discree-ime Le f = f + k where k Z, For F F, A cos(πf + θ) A cos(πf + θ) excep a discree pois i ime. x () = A e j(πf +θ) = A e j(π(f +k)+θ) = A e j(πf +θ) e j(πk) = x () = x () Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 3 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 3 / 5

9 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 33 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 34 / 5 Uiqueess: Discree-ime Harmoically Relaed Complex Expoeials Therefore, ds-ime siusoids are uique for f [, ). For ay siusoid wih f [, ), f [, ) such ha x () = A e j(πf +θ) = A e j(πf +θ) = x (). Example: A ds-ime siusoid wih frequecy f = 4.5 is he same as a ds-ime siusoid wih frequecy f = =.5. Example: A ds-ime siusoid wih frequecy f = 7 is he 8 same as a ds-ime siusoid wih frequecy f = 7 + =. 8 8 Harmoically relaed s k () = e jkω = e jπkf, (cs-ime) k =, ±, ±,... Scieific Desigaio Frequecy (Hz) k for F = 8.7 C- 8.7 C.35 C C C C C Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 35 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 3 / 5

10 Harmoically Relaed Complex Expoeials Scieific Desigaio Frequecy (Hz) k for F = 8.7 C C C C4 (middle C). 3 C C C C Harmoically Relaed Complex Expoeials Wha does he family of harmoically relaed siusoids s k () have i commo? Harmoically relaed s k () = e jkω = e jπ(kf), (cs-ime) k =, ±, ±,... fud. period: T,k = cyclic frequecy = kf period: T k = ay ieger muliple of T commo period: T = k T,k = F C C C3 C4 C5 C C7 C8 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 37 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 38 / 5 Harmoically Relaed Complex Expoeials Harmoically Relaed Complex Expoeials Discree-ime Case: For periodiciy, selec f = N where N Z: Harmoically relaed s k () = e jπkf = e jπk/n, (ds-ime) k =, ±, ±,... There are oly N disic ds-ime harmoics: s k (), k =,,,..., N. s k+n () = e jπ(k+n)/n = e jπk/n e jπn/n = e jπk/n = e jπk/n = s k () Therefore, here are oly N disic ds-ime harmoics: s k (), k =,,,..., N. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 39 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 4 / 5

11 .4 Aalog-o-Digial ad Digial-o-Aalog Coversio.4 Aalog-o-Digial ad Digial-o-Aalog Coversio Aalog-o-Digial Coversio Aalog-o-Digial Coversio A/D coverer A/D coverer x () a Sampler x() Quaizer x () q Coder... x () a Sampler x() Quaizer x () q Coder... Aalog Discree-ime Quaized Digial Aalog Discree-ime Quaized Digial Samplig: coversio from cs-ime o ds-ime by akig samples a discree ime isas E.g., uiform samplig: x() = x a (T ) where T is he samplig period ad Z Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 4 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 4 / 5.4 Aalog-o-Digial ad Digial-o-Aalog Coversio.4 Aalog-o-Digial ad Digial-o-Aalog Coversio Aalog-o-Digial Coversio Aalog-o-Digial Coversio A/D coverer A/D coverer x () a Sampler x() Quaizer x () q Coder... x () a Sampler x() Quaizer x () q Coder... Aalog Discree-ime Quaized Digial Aalog Discree-ime Quaized Digial Quaizaio: coversio from ds-ime cs-valued o a ds-ime ds-valued quaizaio error: e q () = x q () x() for all Z Codig: represeaio of each ds-value x q () by a b-bi biary sequece e.g., if for ay, x q () {,,...,, 7}, he he coder may use he followig mappig o code he quaized ampliude: Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 43 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 44 / 5

12 Aalog-o-Digial Coversio.4 Aalog-o-Digial ad Digial-o-Aalog Coversio Samplig Theorem.4 Aalog-o-Digial ad Digial-o-Aalog Coversio x () a A/D coverer Sampler x() Quaizer x () q Coder... If he highes frequecy coaied i a aalog x a () is F max = B ad he is sampled a a rae Aalog Example coder: Discree-ime Quaized Digial F s > F max = B he x a () ca be exacly recovered from is sample values usig he ierpolaio fucio g() = si(πb) πb Noe: F N = B = F max is called he Nyquis rae. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 45 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 4 / 5.4 Aalog-o-Digial ad Digial-o-Aalog Coversio.4 Aalog-o-Digial ad Digial-o-Aalog Coversio Samplig Theorem Badlimied Ierpolaio Samplig Period = T = F s = Samplig Frequecy Therefore, give he ierpolaio relaio, x a () ca be wrie as x a () = x a () = = = x a (T )g( T ) x() g( T ) where x a (T ) = x(); called badlimied ierpolaio. badlimied ierpolaio fucio -- sic x() samples Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 47 / 5 Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 48 / 5

13 .4 Aalog-o-Digial ad Digial-o-Aalog Coversio.4 Aalog-o-Digial ad Digial-o-Aalog Coversio Digial-o-Aalog Coversio Digial-o-Aalog Coversio origial/badlimied ierpolaed x() origial/badlimied ierpolaed zero-order hold -3T -T -T T T 3T Commo ierpolaio approaches: badlimied ierpolaio, zero-order hold, liear ierpolaio, higher-order ierpolaio echiques, e.g., usig splies I pracice, cheap ierpolaio alog wih a smoohig filer is employed. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 49 / 5 Commo ierpolaio approaches: badlimied ierpolaio, zero-order hold, liear ierpolaio, higher-order ierpolaio echiques, e.g., usig splies I pracice, cheap ierpolaio alog wih a smoohig filer is employed. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 5 / 5.4 Aalog-o-Digial ad Digial-o-Aalog Coversio Digial-o-Aalog Coversio origial/badlimied ierpolaed -3T liear ierpolaio -T -T T T 3T Commo ierpolaio approaches: badlimied ierpolaio, zero-order hold, liear ierpolaio, higher-order ierpolaio echiques, e.g., usig splies I pracice, cheap ierpolaio alog wih a smoohig filer is employed. Professor Deepa Kudur (Uiversiy of Toroo)Iroducio o Digial Sigal Processig 5 / 5