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Signals and Systems Linear System Theory

Signals and Systems Linear System Theory. EE 112 - Lecture Eight Signal classification. Signal. A signal is a pattern of variation that carry information. Signals are represented mathematically as a function of one or more independent variable

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Signals and Systems Linear System Theory

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  1. Signals and SystemsLinear System Theory EE 112 - Lecture Eight Signal classification Khosrow Ghadiri - EE Dept. SJSU

  2. 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. Khosrow Ghadiri - EE Dept. SJSU

  3. 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 • … Khosrow Ghadiri - EE Dept. SJSU

  4. 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 Khosrow Ghadiri - EE Dept. SJSU

  5. A continuous-time signal • Continuous-time signal x(t), the independent variable, t is Continuous-time. The signal itself needs not to be continuous. Khosrow Ghadiri - EE Dept. SJSU

  6. A continuous-time signal • x=0:0.05:5; • y=sin(x.^2); • plot(x,y); Khosrow Ghadiri - EE Dept. SJSU

  7. A piecewise continuous-time signal • A piecewise continuous-time signal Khosrow Ghadiri - EE Dept. SJSU

  8. A discrete-time signal A discrete signal is defined only at discrete instances. Thus, the independent variable has discrete values only. Khosrow Ghadiri - EE Dept. SJSU

  9. Mathlab file • x = 0:0.1:4; • y = sin(x.^2).*exp(-x); • stem(x,y) Khosrow Ghadiri - EE Dept. SJSU

  10. 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. Khosrow Ghadiri - EE Dept. SJSU

  11. A piecewise discrete-time signal • A piecewise discrete-time signal Khosrow Ghadiri - EE Dept. SJSU

  12. 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: Khosrow Ghadiri - EE Dept. SJSU

  13. Power and Energy in a Physical System • The instantaneous power • The total energy • The average power Khosrow Ghadiri - EE Dept. SJSU

  14. 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 Khosrow Ghadiri - EE Dept. SJSU

  15. Power and Energy • Example 1: The signal is given below is energy or power signal. Explain. • This signal is energy signal Khosrow Ghadiri - EE Dept. SJSU

  16. Power and Energy • Example 2: The signal is given below is energy or power signal. Explain. • This signal is energy signal Khosrow Ghadiri - EE Dept. SJSU

  17. 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 Khosrow Ghadiri - EE Dept. SJSU

  18. 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: Khosrow Ghadiri - EE Dept. SJSU

  19. 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 Khosrow Ghadiri - EE Dept. SJSU

  20. 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 Khosrow Ghadiri - EE Dept. SJSU

  21. 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: Khosrow Ghadiri - EE Dept. SJSU

  22. 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 Khosrow Ghadiri - EE Dept. SJSU

  23. 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. Khosrow Ghadiri - EE Dept. SJSU

  24. 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. Khosrow Ghadiri - EE Dept. SJSU

  25. Taylor series and Euler relation • General Taylor series • Expanding sin and cos • Expanding exponential • Euler relation Khosrow Ghadiri - EE Dept. SJSU

  26. 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. Khosrow Ghadiri - EE Dept. SJSU

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