Linear Systems Khosrow Ghadiri - EE Dept. SJSU 1Signals and SystemsLinear System TheoryEE 112 - Lecture Eight Signal classificationSignals and Systems Khosrow Ghadiri - EE Dept. SJSU 2Signal A signal is a pattern of variation that carry information. Signals are represented mathematically as a function of one or more independent variable  A picture is brightness as a function of two spatial variables, x and y. In this course signals involving a single independent variable, generally refer to as a time, t are considered. Although it may not represent time in specific application A signal is a real-valued or scalar-valued function of an independent variable t.Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 3Example of signals Electrical signals like voltages, current and EM field in circuit Acoustic signals like audio or speech signals (analog or digital) Video signals like intensity variation in an image Biological signal like sequence of bases in gene  Noise which will be treated as unwanted signal …Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 4Signal classification Continuous-time and Discrete-time  Energy and Power Real and Complex Periodic and Non-periodic  Analog and Digital Even and Odd  Deterministic and RandomSignals and Systems Khosrow Ghadiri - EE Dept. SJSU 5A continuous-time signal Continuous-time signal x(t), the independent variable, t is Continuous-time. The signal itself needs not to be continuous.0 1 2 3 4 5 6−0.6−0.4−0.200.20.40.60.8Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 6A continuous-time signal x=0:0.05:5;  y=sin(x.^2);  plot(x,y);Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 7A piecewise continuous-time signal A piecewise continuous-time signal0; 0 101; 10 20()1; 20 252; 25 30ttxttt≤≤<≤=−≤≤<≤Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 8A discrete-time signalA discrete signal is defined only at discrete instances. Thus, the independent variable has discrete values only.0 1 2 3 4 5 6−0.6−0.4−0.200.20.40.60.8[]xnSignals and Systems Khosrow Ghadiri - EE Dept. SJSU 9Mathlab file  x = 0:0.1:4;  y = sin(x.^2).*exp(-x); stem(x,y)Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 10Sampling A discrete signal can be derived from a continuous-time signal by sampling it at a uniform rate.  If denotes the sampling period and denotes an integer thatmay assume positive and negative values, Sampling a continuous-time signal x(t) at time yields a sample of value  For convenience, a discrete-time signal is represented by a sequence of numbers: We write Such a sequence of numbers is referred to as a time series.τtnτ=()xnτ[] ( )xn xn nτ=∀nSignals and Systems Khosrow Ghadiri - EE Dept. SJSU 11A piecewise discrete-time signal A piecewise discrete-time signal []0; 0 101; 10 201; 20 252; 25 30nnxnnn≤≤<≤=−≤≤<≤Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 12Energy and Power Signals X(t) is a continuous power signal if: X[n] is a discrete power signal if: X(t) is a continuous energy signal if: X[n] is a discrete energy signal if:()210lim2TTTxt dtT−→∞<<∞∫[]210lim21NNnNxnN→∞=−<<∞+∑()20 xt dt∞−∞<<∞∫[]20nxn∞=−∞<<∞∑Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 13Power and Energy in a Physical System The instantaneous power The total energy The average power() ()() ()21Ptvtit vtR==() ()() ()22 211 121tt ttt tP t dt v t i t dt v t dtR==∫∫ ∫() ()2211221 21111ttttPtdt v tdttt tt R=−−∫∫Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 14Power and Energy By definition, the total energy over the time interval in a continuous-time signal is:denote the magnitude of the (possibly complex) number The time average power By definition, the total energy over the time interval in a discrete-time signal is: The time average power12ttt<<()xt()212ttExtdt=∫()xt()xt()212211ttPxtdttt=−∫21nnn≤≤[]xn[]212nnnExn==∑[]2122111nnnPxnnn==−+∑Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 15Power and Energy Example 1:The signal is given below is energy or power signal.Explain. This signal is energy signal()122011119lim lim 3 lim 9 lim 002222TTTTTTPxtdt dttTTTT−→∞ →∞ →∞ →∞=====∫∫()221013990TTExtdt dtt−====∫∫()xt()xt301tSignals and Systems Khosrow Ghadiri - EE Dept. SJSU 16Power and Energy Example 2:The signal is given below is energy or power signal.Explain. This signal is energy signal[]xn[]2132201127lim lim 3 lim 021 21 21nNNNnnPxnNNN→∞ →∞ →∞=====+++∑∑[]213220327nnnExn====∑∑[]xn301n2Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 17Real and Complex A value of a complex signal is a complex number The complex conjugate, of the signal is; Magnitude or absolute value  Phase or angle ()xt() ()()(){}() (){}()1212Where 1 Re Imxt x t jx tjxt x t xt x t=+=− = =()xt()xt∗()()()(){}() () (){}() ()1211Re Im22xt xt jxtxt xt x t xt xt x tj∗∗∗=− =+ = − () () () () ()221212xt xt xt xtxt∗=+=()()()121tanxt x t x t−=Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 18Periodic and Non-periodic A signal or is a periodic signal ifHere, and are fundamental period, which is the smallest positive values when Example:()xt[]xn() ( )[] [ ]00;, any integer;, any integerxt xt mT t mxn xn mN n m=+ ∀ ==+ ∀ =oToN1m=0() cos( ); ;2oxt A t t fωωθπ=+−∞<<∞=000 000()cos([ ])cos( )2; 0,1,2......xt T A t mT A tTmωθωθπω+= + += +⇒= =Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 19Analog and DigitalDigital signal is discrete-time signal whose values belong to a defined set of real numbers Binary signal is digital signal whose values are 1 or 0 Analog signal is a non-digital signal12{, ,..., }Naa a[] ( ) 0 1;nx n x t or n== ∀[] ( ) ; 1nixn xt a i N== ≤≤Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 20Even and Odd Even SignalsThe continuous-time signal /discrete-time signal is an even signal if it satisfies the condition  Even signals are symmetric about the vertical axis  Odd SignalsThe signal is said to be an odd signal if it satisfies the condition Odd signals are anti-symmetric (asymmetric) about the