Linear SystemsKhosrow Ghadiri - EE Dept. SJSU1 Signals and Systems Linear System Theory EE Lecture Eight Signal classification

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU2 Signal  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 SystemsKhosrow Ghadiri - EE Dept. SJSU3 Example 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 SystemsKhosrow Ghadiri - EE Dept. SJSU4 Signal classification  Continuous-time and Discrete-time  Energy and Power  Real and Complex  Periodic and Non-periodic  Analog and Digital  Even and Odd  Deterministic and Random

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU5 A continuous-time signal  Continuous-time signal x(t), the independent variable, t is Continuous-time. The signal itself needs not to be continuous.

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU6 A continuous-time signal  x=0:0.05:5;  y=sin(x.^2);  plot(x,y);

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU7 A piecewise continuous-time signal  A piecewise continuous-time signal

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU8 A discrete-time signal A discrete signal is defined only at discrete instances. Thus, the independent variable has discrete values only.

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU9 Mathlab file  x = 0:0.1:4;  y = sin(x.^2).*exp(-x);  stem(x,y)

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU10 Sampling  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 that may 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.

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU11 A piecewise discrete-time signal  A piecewise discrete-time signal

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU12 Energy 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:

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU13 Power and Energy in a Physical System  The instantaneous power  The total energy  The average power

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU14 Power 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 power

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU15 Power and Energy  Example 1: The signal is given below is energy or power signal. Explain.  This signal is energy signal

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU16 Power and Energy  Example 2: The signal is given below is energy or power signal. Explain.  This signal is energy signal

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU17 Real 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

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU18 Periodic and Non-periodic  A signal or is a periodic signal if Here, and are fundamental period, which is the smallest positive values when Example:

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU19 Analog and Digital Digital 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 signal

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU20 Even and Odd  Even Signals The continuous-time signal /discrete-time signal is an even signal if it satisfies the condition  Even signals are symmetric about the vertical axis  Odd Signals The signal is said to be an odd signal if it satisfies the condition  Odd signals are anti-symmetric (asymmetric) about the time origin

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU21 Even and Odd signals:Facts  Product of 2 even or 2 odd signals is an even signal  Product of an even and an odd signal is an odd signal  Any signal (continuous and discrete) can be expressed as sum of an even and an odd signal:

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU22 Complex-Valued Signal Symmetry  For a complex-valued signal  is said to be conjugate symmetric if it satisfies the condition  where  is the real part and is the imaginary part;  is the square root of -1

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU23 Deterministic and Random signal  A signal is deterministic whose future values can be predicted accurately.  Example:  A signal is random whose future values can NOT be predicted with complete accuracy  Random signals whose future values can be statistically determined based on the past values are correlated signals.  Random signals whose future values can NOT be statistically determined from past values are uncorrelated signals and are more random than correlated signals.

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU24 Deterministic and Random signal(contd…)  Two ways to describe the randomness of the signal are:  Entropy: This is the natural meaning and mostly used in system performance measurement.  Correlation: This is useful in signal processing by directly using correlation functions.

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU25 Taylor series and Euler relation  General Taylor series  Expanding sin and cos  Expanding exponential  Euler relation

Signals and SystemsKhosrow Ghadiri - EE Dept. SJSU26 Example  The signal and shown below constitute the real and imaginary parts of a complex-valued signal.  What form of symmetry does have?  Answer  is conjugate symmetric.