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1.introduction to signals
- 1. Introduction to Signals and Systems Prof. Satheesh Monikandan.B INDIAN NAVAL ACADEMY (INDIAN NAVY) EZHIMALA sathy24@gmail.com 101 INAC-L-AT19
- 2. • Course Code : PCL402 • Course title : SIGNALAS AND SYSTEMS • Credit Hours : 3.5 • Semester : AT19 • Book 1 : A.Oppenheim, A.Willsky and S.Hamid Nawab, “Signals and Systems”, 2nd Edition, 2013. • Book 2 : Simon Haykin and Barry V.Veen, “Signals and Systems”. • Book 3 : Dr.H.P.Hsu, “Signals and Systems”, Schaum’s Outlines Series. • Book 4 : Luis F.Chaparro and Aydin Akan, “Signals and Systems using MATLAB”.
- 3. Syllabus Contents • Introduction to Signals and Systems • Time-domain Analysis of LTI Systems • Frequency-domain Representations of Signals and Systems • Sampling • Hilbert Transform • Laplace Transform
- 4. Introduction - Signals • Signal - the fundamental quantity of representing some information is called a signal. • Analog Signal - defined with respect to the time. – Human voice • Digital Signal - appropriation of analog signals. – Computer Keyboard • Fourier Series • Fourier Transform • Finite Energy Signals • Periodic Signals • Signal Energy • Signal Power
- 5. Introduction - Systems • System - a mathematical model, a piece of code / software, or a physical device, or a black box whose input is a signal and it performs some processing on that signal, and the output is a signal. • Continuous / Discrete Time Systems
- 6. Introduction – DT Systems • Type of systems whose input and output both are discrete signals or digital signals are called digital systems. • Digital Cameras
- 7. Classifications of Signals • Continuous Time and Discrete Time Signals • Deterministic and Non-deterministic Signals • Even and Odd Signals • Periodic and Aperiodic Signals • Energy and Power Signals • Real and Imaginary Signals
- 8. Classification of Signals Continuous time (CT) and discrete time (DT) signals CT signals take on real or complex values as a function of an independent variable that ranges over the real numbers and are denoted as x(t). DT signals take on real or complex values as a function of an independent variable that ranges over the integers and are denoted as x[n]. Note the subtle use of parentheses and square brackets to distinguish between CT and DT signals.
- 9. Discrete-Time Signals • Sampling is the acquisition of the values of a continuous-time signal at discrete points in time • x(t) is a continuous-time signal, x[n] is a discrete-time signal x x where is the time between sampless sn nT T
- 10. Classifications of Signals • Non-deterministic Signals • Even Signals – A signal is said to be even when it satisfies the condition x(t) = x(-t) – cos t is even function – ƒe (t ) = ½[ƒ(t ) +ƒ(-t )]
- 11. Classifications of Signals • Odd Signals – A signal is said to be odd when it satisfies the condition x(t) = -x(-t) – sin t is odd function – ƒ(t ) = ƒe (t ) + ƒo (t ) • Periodic Signals – A signal is said to be periodic if it satisfies the condition x(t) = x(t + T) or x[n] = x[n + N]
- 12. Classifications of Signals • Energy and Power Signals – A signal is said to be energy signal when it has finite energy. – A signal is said to be power signal when it has finite power. – A signal cannot be both, energy and power simultaneously. – A signal may be neither energy nor power signal. – Power of energy signal = 0 – Energy of power signal = ∞
- 13. Power and Energy Signals • Power Signal – Infinite duration – Normalized power is finite and non- zero – Normalized energy averaged over infinite time is infinite – Mathematically tractable • Energy Signal – Finite duration – Normalized energy is finite and non- zero – Normalized power averaged over infinite time is zero – Physically realizable
- 14. Classifications of Signals • Real and Imaginary Signals – A signal is said to be real when it satisfies the condition x(t) = x*(t) – A signal is said to be odd when it satisfies the condition x(t) = -x*(t) – For a real signal, imaginary part should be zero. – For an imaginary signal, real part should be zero.
- 15. Operations on Signals • In Amplitude and Time • Amplitude Scaling • Addition • Subtraction • Multiplication • Time Shifting • Time Scaling • Time Reversal
- 16. Time Shifting • The original signal x(t) is shifted by an amount tₒ. • X(t)X(t-to) Signal Delayed Shift to the right
- 17. Time Shifting Contd. • X(t)X(t+to) Signal Advanced Shift to the left
- 18. Properties of Signals • Finite vs. Infinite Length – Finite length signals are used when dealing with discrete-time signals – f(t) is a finite-length signal if it is nonzero over a finite interval – f(t) is a infinite-length signal if it is nonzero over all real numbers
- 19. Properties of Signals • Causal vs. Anticausal vs. Noncausal – Causal signals are signals that are zero for all negative time. – Anticausal signals are signals that are zero for all positive time. – Noncausal signals are signals that have nonzero values in both positive and negative time.
- 20. Properties of Signals • Even vs. Odd Symmetry – Even signals can be easily spotted as they are symmetric around the vertical axis. – Any signal can be written as a combination of an even and odd signal. That is, every signal has an odd-even decomposition.
- 21. Properties of Signals • Deterministic vs. Random (Periodicity) – Future values of deterministic signals can be calculated from past values with complete confidence. – Future values of a random signal cannot be accurately predicted and can usually only be guessed based on the averages of sets of signals.
- 22. Properties of Signals • Bounded Signals • Unbounded Signals • Absolutely Integrable Signals • Absolutely Summable Signals • Stable Signals • Causal Signals • Reflection property • Right-handed Signals • Left-handed Signals • Two-sided Signals
- 23. Elementary Signals • Unit impulse signal • Unit sample sequence • Unit step signal • Unit step sequence • Ramp signal • Ramp sequence • Exponential signal • Exponential sequence • Sinusoidal signal • Real Exponential signal (C and α are real) • Growing Exponential signal • Decaying Exponential signal • Complex Exponential signal
- 24. Elementary Signals Sinusoidal & Exponential Signals • Sinusoids and exponentials are important in signal and system analysis because they arise naturally in the solutions of the differential equations. • Sinusoidal Signals can expressed in either of two ways : cyclic frequency form- A sin 2Пfot = A sin(2П/To)t radian frequency form- A sin ωot ωo = 2Пfo = 2П/To To = Time Period of the Sinusoidal Wave
- 25. Sinusoidal & Exponential Signals Contd. x(t) = A sin (2Пfot+ θ) = A sin (ωot+ θ) x(t) = Aeat Real Exponential = Aejωωt = A[cos (ωot) +j sin (ωot)] Complex Exponential θ = Phase of sinusoidal wave A = amplitude of a sinusoidal or exponential signal fo = fundamental cyclic frequency of sinusoidal signal ωo = radian frequency Sinusoidal signal
- 26. Discrete Time Exponential and Sinusoidal Signals • DT signals can be defined in a manner analogous to their continuous-time counter part x[n] = A sin (2Пn/No+θ) = A sin (2ПFon+ θ) x[n] = an n = the discrete time A = amplitude θ = phase shifting radians, No = Discrete Period of the wave 1/N0 = Fo = Ωo/2 П = Discrete Frequency Discrete Time Sinusoidal Signal Discrete Time Exponential Signal
- 27. Introduction to Systems • Def1: A system is defined mathematically as a unique operator or transformation that maps an input signal in to an output signal. • Def2: A system is any process that produces an output signal in response to an input signal. • The input is known as excitation and the output is known as response. • Examples – Removal of Noise » Sharpening the Image » Removal of echoes in an audio recording » Reducing crosstalk in telephony » Equalizer » Radar » Sonar
- 28. Classification of Systems • Linear and Non-linear Systems - said to be linear when it satisfies superposition and homogeneity principles. – Response of overall system is equal to response of individual system. • Time Variant and Time Invariant Systems - said to be time variant if its input and output characteristics vary with time. • Linear Time variant and Linear Time Invariant (LTI) systems • Static/memoryless and Dynamic/memory Systems • Causal and Non-causal Systems • Invertible and Non-Invertible Systems • Stable and Unstable Systems
- 29. Classification of Systems • Causal Systems - output depends upon present and past inputs, and does not depend upon future input. • Non-causal Systems - output depends upon future inputs also. • Invertible and Non-Invertible Systems - input of the system appears at the output. • Continuous-time (CT) and Discrete-time(DT) systems • Stable and Unstable Systems - said to be stable only when the output is bounded for bounded input. – For a bounded signal, amplitude is finite. – Squaring unit step bounded input (stable) – Integrating unit step bounded input (unstable)
- 30. Properties of Systems • Linearity • Memory • Time/shift Invariance • Stability • Causality • Invertibility
- 31. Properties of Systems • Time/shift Invariance • A system is time invariant if a shift in the time domain corresponds to the same shift in the output. • Stability • BIBO
- 32. Properties of Systems • Causality - y[n] = x[n] + x[n-2] » y[n] = x[n-1] – x[n-3] » y[n] = 7x[n-5] » Practically realizable » Real-time systems are causal systems » Causal system is memory less system. » Condition for causality: h(t) = 0 for t<0 » Non-causal system is practically not realizable. » Population growth » Weather forecasting » Planning commission
- 33. Sampling of Signals • Sampling is the reduction of a CT signal into a DT signal. • The value of the signal is measured at certain intervals in time. • A sampler is a subsystem or operation that extracts samples from a continuous signal.
- 34. Sampling Theorem • A continuous time signal can be represented in its samples and can be recovered back when sampling frequency (fs ) is greater than or equal to the twice the highest frequency component (fm ) of message signal. • The rate at which samples of an analog signal are taken in order to be converted into digital form is called sampling rate. • If the sampling rate is less than twice the input frequency, the output frequency will be different from the input which is known as aliasing. • Aliasing can be referred to as “the phenomenon of a high-frequency component in the spectrum of a signal, taking on the identity of a low-frequency component in the spectrum of its sampled version.”
- 37. Sampling of Signals • The process of measuring the instantaneous values of continuous-time signal in a discrete form.
- 38. Sampling of Signals • To discretize the signals, the gap between the samples should be fixed. That gap can be termed as a sampling period (Ts ). • Sampling frequency (sampling rate) is the reciprocal of the sampling period. • The sampling rate denotes the number of samples taken per second, or for a finite set of values. • The rate of sampling should be such that the data in the message signal should neither be lost nor it should get over-lapped. • Used for digitization of analog signals and processing signals in computers. • An audio CD can represent frequencies up to 22.05 KHz, where the Nyquist frequency is 44.1 Khz.
- 39. Signal Examples • Electrical signals --- voltages and currents in a circuit • Acoustic signals --- audio or speech signals (analog or digital) • Video signals --- intensity variations in an image (e.g. a CT scan) • Biological signals --- sequence of bases in a gene • Noise: unwanted signal :
- 41. Definitions • Voltage – the force which moves an electrical current against resistance • Waveform – the shape of the signal (previous slide is a sine wave) derived from its amplitude and frequency over a fixed time (other waveform is the square wave) • Amplitude – the maximum value of a signal, measured from its average state • Frequency (pitch) – the number of cycles produced in a second – Hertz (Hz). Relate this to the speed of a processor eg 1.4GigaHertz or 1.4 billion cycles per second
- 42. Analog or Digital • Analog Message: continuous in amplitude and over time – AM, FM for voice sound – Traditional TV for analog video – First generation cellular phone (analog mode) – Record player • Digital message: 0 or 1, or discrete value – VCD, DVD – 2G/3G cellular phone – Data on your disk
- 43. Representation of Impulse Function The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse.
- 44. The Unit Rectangle Function The unit rectangle or gate signal can be represented as combination of two shifted unit step signals as shown
- 45. The Unit Triangle Function A triangular pulse whose height and area are both one but its base width is not, is called unit triangle function. The unit triangle is related to the unit rectangle through an operation called convolution.
- 46. Sinc Function sin sinc t t t