Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar
Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor E.C.; Jervis B.W.; Digial Signal Processing, Pearson Educaion. ISBN: 978-8-37-08-8
Signal A signal is defined as a funcion of one or more variables which conveys informaion on he naure of a physical phenomenon. The value of he funcion can be a real valued scalar quaniy, a complex valued quaniy, or perhaps a vecor. Example: Speech signals, Biomedical signals, raffic signals ec. 3
Sysem Asysemisdefinedasaneniyhamanipulaesone or more signals o accomplish a funcion, hereby yielding new signals. A sysem may be a piece of hardware or sofware, a combinaion of several componens, an equaion or an algorihm. Example: Your mobile se, an FM receiver, your compuer ec.
Signal Classificaion Coninuous Time Signal: A signal x() is said o be a coninuous ime signal if i is definedforallime. y() 5
Discree Time Signal: A discree ime signal x(n)has values specified only a discree poins in ime. x[n] 0 3 5 n 6
Signal Processing A sysem characerized by he ype of operaion ha i performs on he signal. For example, if he operaion is linear, he sysem is called linear. If he operaion is non-linear, he sysem is said o be nonlinear, and so forh. Such operaions are usually referred o as Signal Processing. 7
Basic Elemens of a Signal Processing Sysem Analog inpu signal Analog Signal Processor Analog Signal Processing Analog oupu signal Analog inpu signal AD converer Digial Signal Processor DA converer Analog oupu signal Digial Signal Processing 8
Advanages of Digial over Analogue Signal Processing The same digial machine(hardware) can be used for implemening differen versions of a signal processing operaion of ineres(e.g. filering) merely by making changes o he sofware (program) read ino he machine. On he oher hand, in he case of an analogue machine, he sysem has o be redesigned every ime he signal processing specificaions are changed. DSP is very suiable when a ask is repeaed again and ag.ain 9
Advanages of Digial over Analogue Signal Processing DSP operaions are more accurae han heir analogue counerpars. Digial signals can easily be sored on magneic media ec. In many cases, digial implemenaion of he signal processing sysem is cheaper han is analogue counerpars. 0
Deerminisic Signals: A deerminisic signal behaves in a fixed known way wih respec o ime. Thus, i can be modeled by a known funcion of ime for coninuous ime signals, or a known funcion of a sampler number n, and sampling spacing T for discree ime signals. Random or Sochasic Signals: In many pracical siuaions, here are signals ha eiher canno be described o any reasonable degree of accuracy by explici mahemaical formulas, or such a descripion isoocomplicaedobeofanypracicaluse.thelack of such a relaionship implies ha such signals evolve in ime in an unpredicable manner. We refer o hese signals as random.
Even and Odd Signals Aconinuousimesignalx()issaidoanevensignalifisaisfies he condiion x(-)x()forall The signal x() is said o be an odd signal if i saisfies he condiion x(-)-x() In oher words, even signals are symmeric abou he verical axis or ime origin, whereas odd signals are anisymmeric abou he ime origin. Similar remarks apply o discree-ime signals. even odd odd
Le x e () be he even par and x 0 () be he odd par of a signal x(). Show ha (a) x e () ½ [x() x(-)] and (b) x 0 () ½ [x() x(-)] Proof: x() x e () x 0 ().() Replacing by we ge x(-) x e (-) x 0 (-) or x(-) x e ()-x 0 ()..() Adding () and (), we ge x() x(-) x e () or x e () ½ [x() x(-)] Home work: Prove (b) 3
Problem Find even and odd componens of each of he following signals: (i) x() cos( ) sin() cos()sin() (ii) x() 5 3 9. (iii)x() cos() sin() 3 sin()cos() (iv)x() ( 3 )cos 3 (0)
Periodic Signals A coninuous ime signal x() is periodic if and only if here exiss a T>0 such ha x( T) x() where T is he period ofhe signalin unisofime. ft ishe frequency of he signal in Hz. T is he angular frequency in radians per second. Example: 0 0. 0. Time Period T of his signal 0. sec. Frequency f T 0. 5 Hz 0radians per sec. 5
Coninuous Time Sinusoidal Signal A coninuous ime sinusoidal signal (eiher sine or cosine) is described as x()acos(θ) where f (f is frequency in Hz) and θ is he phase angle in radians or degrees. A is ampliude of he signal. A Tf 6
Discree Time Sinusoidal Signal A discree ime sinusoidal signal may be expressed as x(n)acos(wn θ) - <n< 0-0 6 8 0 7
Energy and Power Signals Theoalenergyofaconinuousimesignalx()isdefinedas E x lim T T T x ()d x ( )d andisaveragepoweris In he case of a discree ime signal x(n), he oal energy ofhe signalis P x lim T T T x ( ) d In he case of a discree ime signal x[nt], he oal energy of he signal is n And is average power is defined by P E dx dx lim N T x N [ n ] N n N x [ ] n 8
Energy and Power Signals A signal is referred o as an energy signal, if and only if he oal energy of he signal saisfies he condiion 0<E< On he oher hand, i is referred o as a power signal, if and only if he average power of he signal saisfies he condiion 0<P< An energy signal has zero average power, whereas a power signal has infinie energy. Periodic signals are usually viewed as power signals, whereas signals ha are non-periodic are energy signals. 9
Whaisheoalenergyofherecangularwaveshown inhefig.? Soluion: E T x ()d T A d x A T TheraisedcosinepulseshowninheFig.isdefinedas Deermine he oal energy of x(). [ ] cos( ) T x() 0 oherwise A -T 0 T 0
[ ] d cos ()d x E x ( ) [ ]d cos ) ( cos d ) cos( cos d 3 cos cos d ( )d 3 cos cos 8 3 sin sin 8 3
Deermine he oal energy of he following signal 5 x( ) 5 0 Soluion: E 5 ( 5) d 5 oherwise 5 5 () d x 3 ( 5) 3 5 (5 ) 3 3 5 (5 ) d 6 3
Find he oal energy of he discree ime signal shown below: Soluion: E [ n ] 3 dx x n Compue he signal energy for x(n) (-0.5) n u(n). Soluion: E n n dx x(n) ( 0.5) (0.5) n n 0 n 0 0.5 0.5 (0.5) 0.75.33 (0.5) 3... 3
Wha is he average power of he riangular waveshown inhefig. Soluion: T T T Px x ()d () d ( ) d T T T 0 T Wha is he average power of he riangular wave shown inhefig. - 0. 0. 0. 0.6 0.8 Soluion: Mahemaically, he above signal may be represened as 0 0 0. x() 0 0. 0. - T
The average power of he signal is P x T 0. x ()d (0 ) d T 0. T 0 0. 0. ( 0 3) d 0. 5 (00 0 )d 0 0. 0. (00 0 9)d 3 P Wha is he average power of y(n) e j3n u(n). dx lim N N N n N y(n) lim N N N n 0 e j3n N N lim lim N n 0 N N n 0 lim (N ) N N N lim N N N N 5