Sigpro 313: Signals, Spectra, and Signal Processing (PDF)

SIGPRO 313: Signals, Spectra and Signal Processing INTRODUCTION Engr. Enmar T. Tuazon Outline  Signal Definition  Signal Classifications  Signal Operations SIGNAL  It is a single-valued function of time that conveys information. Generally, it’s a physical quantity that varies with...

SIGPRO 313: Signals, Spectra and Signal Processing INTRODUCTION Engr. Enmar T. Tuazon Outline  Signal Definition  Signal Classifications  Signal Operations SIGNAL  It is a single-valued function of time that conveys information. Generally, it’s a physical quantity that varies with time, space or any other independent variables. Signal generation is usually associated with system that responds to a stimulus or force.  Mathematically, signal can be described as a function of one or more dependent variables SIGNAL A signal is a mathematical function that is used to represent any fluctuation in our world.  Like our voice, the EM waves, ECG, Noise and more.  We’ll study how to alter signals. SIGNAL, WHY???  Because if we combine these signals, we can create a new signal different from the original.  And in doing so, we might come across a signal that has a better property than the original.  EXAMPLE:  a. S1(t) = 9t - linear  b. S2(t) = 318t² - quadratic Describe two signals : one that vary linearly with time t and the other vary quadratically with t  c. S(x, y) = 4x² + 3xy + 16y² coordinates in plane Signal that describes two independent variables x,y that could represent two spatial coordinates in the plane  d. Cases where functional relationships are unknown or highly complicated Example:  Speech signal can not be described functionally using expressions it can be represented as sum of several sinusoids of different amplitudes, frequencies and phases Speech Signal  A signal is a description of how one parameter varies with another parameter. An example is a voltage changing over time in an electronic circuit, or brightness varying with distance in an image. Continuous signals are usually represented with parentheses, while discrete signals use brackets. All signals use lower case letters, reserving the upper case for the frequency domain. Unless there is a better name available, the input signal is called: x(t) or x[n], while the output is called: y(t) or y[n]. Classification of Signals  Continuous-Time vs Discrete-Time  Analog vs Digital  Periodic vs Aperiodic  Finite vs Infinite  Causal vs Noncausal vs Anticausal  Even or Odd  Deterministic vs Random Continuous-Time vs Discrete- Time Depends on this axis Analog vs Digital Periodic vs Aperiodic A Periodic Signal repeats at some period T.  An Aperiodic Signal do not repeat. Finite vs Infinite Causal vs Noncausal vs Anticausal Even vs Odd Even vs Odd Deterministic vs Random Example Basic Signal Operation Basic Signal Operation  Addition  Subtraction  Multiplication  Differentiation  Integration Addition Subtraction Subtraction Multiplication Multiplication Differentiation Integration Signal Transformation  Time Shifting  Time Scaling  Time Reversal/Flipping Time Shifting A quantity is added to the time parameter in order to advance or delay the signal  This operation just results in the change of position of the signal without affecting its amplitude and span. Delay Advance Time Scaling  Time Shifting is addition and subtraction in the independent variable,  Scaling is multiplication  If the quantity is greater than one, the signal becomes narrower and the operation is called compression.  If the quantity is less than one the signal becomes wider and is called dilation. Time Reversal/Flipping  Thisoperation is the reversal of the time axis, or flipping the signal over the y-axis. SYSTEMS A system is a transformation from one signal (called the input) to another signal (called the output or the response) A system is any process that produces an output signal in response to an input signal. It may: Physical device that perform operations on the signal Example: o Noise filter for speech o When signal passed through the system the signal is processed(filtered) System definition include not only physical devices but also software realization of operations on signal o The operations performed on signal consist of number of mathematical equations (programs) o Provide more flexibility TYPES OF DOMAIN 1. TIME DOMAIN It simply means that the independent variable is measured in time. 2. FREQUENCY DOMAIN The frequency domain is exactly the same as the time domain description, except that the frequency becomes the independent variable. As you know, the frequency describes how fast (how often) something occurs. 3. SPATIAL DOMAIN The spatial domain is exactly the same as the time domain description, except that the spatial becomes the independent variable. Spatial typically describes a distance. SPECTRA  describes the frequency content of the signal SIGNAL PROCESSING  is extracting information from a signal, conditioning signal for subsequent use, signal transformation, or altering a signal structure. ELEMENTS OF SIGNAL PROCESSING Most of signals are analog in nature. The signals are functions of continuous variable as time or space. They usually take values on continuous range. 1. ANALOG/DIRECT SIGNAL PROCESSING  Signals are processed directly by appropriate analog systems for the purpose of changing their characteristics or extract information  Examples: filters, frequency analyzers, frequency multipliers  (Both input and output are in analog form) 1. ANALOG/DIRECT SIGNAL PROCESSING Analog Signal Processing 2. DIGITAL SIGNAL PROCESSING  is the study of signals in a digital representation and the processing methods of these signals. The algorithms required for DSP are sometimes performed using specialized computers, which make use of specialized microprocessors called digital signal processors. 2. DIGITAL SIGNAL PROCESSING  Digital Signal Processing is distinguished from other areas in computer science by the unique type of data it uses: signals. In most cases, these signals originate as sensory data from the real world: seismic vibrations, visual images, sound waves, etc. DSP is the mathematics, the algorithms, and the techniques used to manipulate these signals after they have been converted into a digital form. This includes a wide variety of goals, such as: enhancement of visual images, recognition and generation of speech, compression of data for storage and transmission, etc. 2. DIGITAL SIGNAL PROCESSING Digital Signal Processing 2. DIGITAL SIGNAL PROCESSING  There is a need for interface between analog system and digital processor “Digital to Analog Converter” (A/D)  The Output of A/D converter is a digital signal appropriate as input for digital processor o Digital signal processor o Large programmable digital computer o Small microprocessor designed for specified purpose MAIN APPLICATIONS OF DSP 1. audio signal processing 2. audio and video compression 3. digital image processing 4. speech processing and speech recognition 5. digital communication  Most DSP techniques are based on a divide-and- conquer strategy called superposition. The signal being processed is broken into simple components, each component is processed individually, and the results reunited. This approach has the tremendous power of breaking a single complicated problem into many easy ones. Superposition can only be used with linear systems, a term meaning that certain mathematical rules apply. What it means for a system to be linear, various ways for breaking signals into simpler components, and how superposition provides a variety of signal processing techniques ADVANTAGE OF DSP OVER ANALOG: 1. Flexibility - easy to reprogram/ reconfigure 2. Accuracy - more accurate than analog because of its digital in nature 3. Cheaper - hardware cost is cheaper due its flexibility 4. Easy storage - Results can be transportable and can be stored to be reprocessed offline, allows for the implementation of a more sophisticated signal processing algorithms LIMITATIONS: 1. speed of operation 2. bandwidth considerations CLASSIFICATION OF SIGNALS: 1. ACCORDING TO CHANNEL (a) SINGLE CHANNEL - generated by single source (b) MULTIPLE CHANNEL - generated by multiple sources or multiple sensors CLASSIFICATION OF SIGNALS: 2. ACCORDING TO DIMENSION (a) ONE DIMENSIONAL - a function of single or independent variable (b) MULTI-DIMENSIONAL (M DIMENSIONAL) - has M-dependent variables Example: Classify the following signals according to its channel and dimension: 1. S(t) = A sin 3 t 𝑠1 (𝑡) 2. 𝑠 𝑡 = 𝑠2 (𝑡) 𝑠3 (𝑡) 3. S(t) = t² + 4t – 8 4. S(x, y) = 4x² + 9xy – 13y³ Example: 5. Consider a video signal (colored) y x  𝐼𝑟 (𝑥, 𝑦, 𝑡) 𝐼 𝑥, 𝑦, 𝑡 = 𝐼𝑔 (𝑥, 𝑦, 𝑡) 𝐼𝑏 (𝑥, 𝑦, 𝑡) CLASSIFICATION OF SIGNALS: 3. ACCORDING TO THE SIGNAL VALUE a. Real b. Complex CLASSIFICATION OF SIGNALS: 4. CLASSIFICATION ACCORDING TO CHARACTERISTICS OF TIME VARIABLE AND THE VALUE THEY TAKE (a) CONTINUOUS–TIME SIGNALS OR ANALOG SIGNALS - are defined for every value of time they take on continuous interval (a, b), where a can be -∞ and b can be + ∞. - Can be described mathematically by functions of continuous variables Example a. speech b. x1(t) = cosπt c. x2(t) = e -|t| , - ˂ t ˃  CLASSIFICATION OF SIGNALS: 4. CLASSIFICATION ACCORDING TO CHARACTERISTICS OF TIME VARIABLE AND THE VALUE THEY TAKE (b) DISCRETE–TIME SIGNALS - are defined only at certain specific values of time. These time instants need not to be equidistant, but they are often equally spaced intervals. Example a. Function of integer variable 𝑥 𝑡𝑛 = 𝑒 − 𝑡𝑛 , 𝑛 = 0, ±1, ±2,... b. It can also be represented by complex or real numbers c. To emphasize the discrete-time nature of the system the signal can be denoted as x(n) instead of x(t) DERIVATION OF DISCRETE TIME SIGNALS  Byselecting values of an analog signal at discrete time instants (sampling).  Byaccumulating a variable over a period of time CLASSIFICATION OF SIGNALS: 4. CLASSIFICATION ACCORDING TO CHARACTERISTICS OF TIME VARIABLE AND THE VALUE THEY TAKE (c) CONTINUOUS–VALUED SIGNALS- if a signal takes on all possible values on infinite or infinite range. CLASSIFICATION OF SIGNALS: 4. CLASSIFICATION ACCORDING TO CHARACTERISTICS OF TIME VARIABLE AND THE VALUE THEY TAKE (d) DISCRETE–VALUED SIGNALS - if the signal takes on values from a finite set of possible values. NOTE: A discrete-time and discrete-valued signal refers to digital signal. EXAMPLE: Identify the following signals according to the time and valued they take. SIGNAL MODEL: 1. DETERMINISTIC SIGNAL - a signal that can be uniquely described by an explicit mathematical expressions, table of data, or a well-defined rule. All past, present and future values of the signal are known precisely without uncertainty. SIGNAL MODEL: 2. RANDOM SIGNAL - signals that cannot be described by explicit mathematical formulas. Random signal