# Digital signal processing transform analysis of lti systems

For analysis of continuous time LTI system Laplace transform is used. And for analysis of discrete time LTI system z transform is used. Z transform is mathematical tool used for conversion of time domain into frequency domain z domain and is a function of the complex valued variable Z.

The z transform of a discrete time signal x n denoted by. X z and given as. But it is useful for values of z for which sum is finite. The DFT can be determined by evaluating z transform. Z transform is widely used for analysis and synthesis of digital filter. Z transform is used for linear filtering.

ROC is going to decide whether system is stable or unstable. ROC decides the type of sequences causal or anti-causal. ROC also decides finite or infinite duration sequences. Fig show the plot of z transforms. The z transform has real and imaginary parts. Thus a plot of imaginary part versus real part is called complex z-plane. The radius of circle is 1 called as unit circle. This complex z plane is used to show ROC, poles and zeros. Q Determine z transform of following signals. Also draw ROC. Q Find linear convolution using z transform.

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The linearity property states that if z. The Time shifting property states that if z x n. This property states that if. The Time reversal property states that if z. The Differentiation property states that if z. The Circular property states that if z. Convolution of two sequences in time domain corresponds to multiplication of its Z transform sequence in frequency domain. The Correlation of two sequences states that if z.

Initial value theorem states that if z.

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Final value theorem states that if z. There is a close relationship between Z transform and Fourier transform. Z transform of sequence x n is given by. Fourier transform of sequence x n is given by. Thus we can be written as.

### IOE Syllabus – Digital Signal Analysis and Processing DSAP

Thus, X z can be interpreted as Fourier Transform of signal sequence x n r —n. The signal can be converted from time domain into z domain with the help of z transform ZT.Discrete time signals and systems [8 hours] 1.

Discrete time signal, basic signal types 1. Energy signal, power signal 1. Periodicity of discrete time signal 1. Transformation of independent variable 1. Discrete time Fourier series and properties 1. Discrete time Fourier transform and properties 1.

Discrete time system properties 1. Frequency response of LTI system 1. Sampling of continuous time signal, spectral properties of sampled signal. Z-transform [4 hours] 2. Defintion, convergence of Z-transform and region of convergence 2. Properties of Z-transform linearity, time shift, multiplication by exponential sequence, differentiation, time reversal, convolution, multiplication 2.

Inverse z-transform by long division and partial fraction expansion. Analysis of LTI system in frequency domain [6 hours] 3. Frequency response of LTI system, response to complex exponential 3. Linear constant co-efficient difference equation and corresponding system function 3. Relationship of frequency response to pole-zero of system 3.

Linear phase of LTI system and its relationship to causality. Discrete filter structures [8 hours] 4. Quantization effect truncation, roundinglimit cycles and scaling.Signals and systems is an aspect of electrical engineering that applies mathematical concepts to the creation of product design, such as cell phones and automobile cruise control systems.

Absorbing the core concepts of signals and systems requires a firm grasp on their properties and classifications; a solid knowledge of algebra, trigonometry, complex arithmetic, calculus of one variable; and familiarity with linear constant coefficient LCC differential equations. The study of signals and systems establishes a mathematical formalism for analyzing, modeling, and simulating electrical systems in the time, frequency, and s — or z — domains.

Signals exist naturally and are also created by people. Some operate continuously known as continuous-time signals ; others are active at specific instants of time and are called discrete-time signals.

Signals pass through systems to be modified or enhanced in some way. Systems that operate on signals are also categorized as continuous- or discrete-time. Mathematics plays a central role in all facets of signals and systems. Specifically, complex arithmetic, trigonometry, and geometry are mainstays of this dynamic and ahem electrifying field of work and study. This article highlights the most applicable concepts from each of these areas of math for signals and systems work.

Here are some of the most important complex arithmetic operations and formulas that relate to signals and systems. Signals — both continuous-time signals and their discrete-time counterparts — are categorized according to certain properties, such as deterministic or random, periodic or aperiodic, power or energy, and even or odd.

But wait! Signals can also be categorized as exponential, sinusoidal, or a special sequence. The unit sample sequence and the unit step sequence are special signals of interest in discrete-time. All the continuous-time signal classifications have discrete-time counterparts, except singularity functions, which appear in continuous-time only. Defining special signals that serve as building blocks for more complex signals makes the creation of custom signal models to suit your needs more systematic and convenient.

Part of learning about signals and systems is that systems are identified according to certain properties they exhibit. Have a look at the core system classifications:. Linearity: A linear combination of individually obtained outputs is equivalent to the output obtained by the system operating on the corresponding linear combination of inputs.

A present input produces the same response as it does in the future, less the time shift factor between the present and future. Memoryless: If the present system output depends only on the present input, the system is memoryless. Causal: The present system output depends at most on the present and past inputs. Stable: A system is bounded-input bound-output BIBO stable if all bounded inputs produce a bounded output.

This table presents core linear time invariant LTI system properties for both continuous and discrete-time systems. Both signals and systems can be analyzed in the time- frequency- and s — and z — domains.

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Leaving the time-domain requires a transform and then an inverse transform to return to the time-domain. As you work to and from the time domain, referencing tables of both transform theorems and transform pairs can speed your progress and make the work easier.

Use this table of common pairs for the continuous-time Fourier transform, discrete-time Fourier transform, the Laplace transform, and the z -transform as needed. Working in the frequency domain means you are working with Fourier transform and discrete-time Fourier transform — in the s -domain.Digital signal processing DSP is the use of digital processingsuch as by computers or more specialized digital signal processorsto perform a wide variety of signal processing operations.

The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronicsa digital signal is represented as a pulse train  which is typically generated by the switching of a transistor.

Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include audio and speech processingsonarradar and other sensor array processing, spectral density estimationstatistical signal processingdigital image processingdata compressionvideo codingaudio codingimage compressionsignal processing for telecommunicationscontrol systemsbiomedical engineeringand seismologyamong others.

DSP can involve linear or nonlinear operations. Nonlinear signal processing is closely related to nonlinear system identification  and can be implemented in the timefrequencyand spatio-temporal domains. The application of digital computation to signal processing allows for many advantages over analog processing in many applications, such as error detection and correction in transmission as well as data compression.

To digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter ADC.

Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a single measurement of amplitude. Quantization means each amplitude measurement is approximated by a value from a finite set.

Rounding real numbers to integers is an example. The Nyquist—Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal. In practice, the sampling frequency is often significantly higher than twice the Nyquist frequency. Theoretical DSP analyses and derivations are typically performed on discrete-time signal models with no amplitude inaccuracies quantization error"created" by the abstract process of sampling. Numerical methods require a quantized signal, such as those produced by an ADC. The processed result might be a frequency spectrum or a set of statistics. But often it is another quantized signal that is converted back to analog form by a digital-to-analog converter DAC. In DSP, engineers usually study digital signals in one of the following domains: time domain one-dimensional signalsspatial domain multidimensional signalsfrequency domainand wavelet domains.

They choose the domain in which to process a signal by making an informed assumption or by trying different possibilities as to which domain best represents the essential characteristics of the signal and the processing to be applied to it. A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain representation.

Time domain refers to the analysis of signals with respect to time. Similarly, space domain refers to the analysis of signals with respect to position, e.

The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Digital filtering generally consists of some linear transformation of a number of surrounding samples around the current sample of the input or output signal. The surrounding samples may be identified with respect to time or space. The output of a linear digital filter to any given input may be calculated by convolving the input signal with an impulse response.

Signals are converted from time or space domain to the frequency domain usually through use of the Fourier transform. The Fourier transform converts the time or space information to a magnitude and phase component of each frequency. With some applications, how the phase varies with frequency can be a significant consideration. Where phase is unimportant, often the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared.

The most common purpose for analysis of signals in the frequency domain is analysis of signal properties. The engineer can study the spectrum to determine which frequencies are present in the input signal and which are missing. Frequency domain analysis is also called spectrum- or spectral analysis. Filtering, particularly in non-realtime work can also be achieved in the frequency domain, applying the filter and then converting back to the time domain.

This can be an efficient implementation and can give essentially any filter response including excellent approximations to brickwall filters. There are some commonly used frequency domain transformations.Account Options Sign in.

DSP Lecture 2: Linear, time-invariant systems

Top charts. New releases. Add to Wishlist. This is the best application to remain updated on your fav. Track your learning, set reminders, edit, add favourite topics, share the topics on social media.

Some of topics Covered in this application are: 1. Introduction to Signals and Systems 2. The Roots of DSP 7. Telecommunications 8. Audio Processing 9. Echo Location Image Processing Signal and Graph Terminology Signal vs. Underlying Process The Normal Distribution Digital Noise Generation Precision and Accuracy Introduction to Linear Time-Invariant Systems Quantization Digital-to-Analog Conversion The Sampling Theorem Selecting the Antialias Filter Analog Filters for Data Conversion Multirate Data Conversion Single Bit Data Conversion In system analysisamong other fields of study, a linear time-invariant system or "LTI system" is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance ; these terms are briefly defined below.

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What's more, there are systematic methods for solving any such system determining h twhereas systems not meeting both properties are generally more difficult or impossible to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers. Linear time-invariant system theory is also used in image processing and field theory [ dubious — discuss ]where the LTI systems have spatial dimensions instead of, or in addition to, a temporal dimension.

## Linear time-invariant system

These systems may be referred to as linear translation-invariant to give the terminology the most general reach. In the case of generic discrete-time i. LTI system theory is an area of applied mathematics which has direct applications in electrical circuit analysis and designsignal processing and filter designcontrol theorymechanical engineeringimage processingthe design of measuring instruments of many sorts, NMR spectroscopy [ citation needed ]and many other technical areas where systems of ordinary differential equations present themselves.

The fundamental result in LTI system theory is that any LTI system can be characterized entirely by a single function called the system's impulse response. The output of the system y t is simply the convolution of the input to the system x t with the system's impulse response h t.

This is called a continuous time system. LTI systems can also be characterized in the frequency domain by the system's transfer functionwhich is the Laplace transform of the system's impulse response or Z transform in the case of discrete-time systems.

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As a result of the properties of these transforms, the output of the system in the frequency domain is the product of the transfer function and the transform of the input. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain. For all LTI systems, the eigenfunctionsand the basis functions of the transforms, are complex exponentials. Since sinusoids are a sum of complex exponentials with complex-conjugate frequencies, if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phasebut always with the same frequency upon reaching steady-state.

LTI systems cannot produce frequency components that are not in the input. LTI system theory is good at describing many important systems. Any system that can be modeled as a linear differential equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up of resistorsinductorsand capacitors RLC circuits. Most LTI system concepts are similar between the continuous-time and discrete-time linear shift-invariant cases.

In image processing, the time variable is replaced with two space variables, and the notion of time invariance is replaced by two-dimensional shift invariance. When analyzing filter banks and MIMO systems, it is often useful to consider vectors of signals. A linear system that is not time-invariant can be solved using other approaches such as the Green function method. The same method must be used when the initial conditions of the problem are not null. The behavior of a linear, continuous-time, time-invariant system with input signal x t and output signal y t is described by the convolution integral : .

And in general, every value of the output can depend on every value of the input. This concept is represented by :. The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse responsescombined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral.Shop now. Click to view larger image. Digital signal processing has never been more prevalent or easier to perform.

It wasn't that long ago when the fast Fourier transform FFTa topic we'll discuss in Chapter 4, was a mysterious mathematical process used only in industrial research centers and universities. Now, amazingly, the FFT is readily available to us all. It's even a built-in function provided by inexpensive spreadsheet software for home computers.

The availability of more sophisticated commercial signal processing software now allows us to analyze and develop complicated signal processing applications rapidly and reliably. We can perform spectral analysis, design digital filters, develop voice recognition, data communication, and image compression processes using software that's interactive both in the way algorithms are defined and how the resulting data are graphically displayed.

Since the mids the same integrated circuit technology that led to affordable home computers has produced powerful and inexpensive hardware development systems on which to implement our digital signal processing designs. The purpose of this book is to build that foundation. In this chapter we'll set the stage for the topics we'll study throughout the remainder of this book by defining the terminology used in digital signal processing, illustrating the various ways of graphically representing discrete signals, establishing the notation used to describe sequences of data values, presenting the symbols used to depict signal processing operations, and briefly introducing the concept of a linear discrete system.

In general, the term signal processing refers to the science of analyzing time-varying physical processes. As such, signal processing is divided into two categories, analog signal processing and digital signal processing. The term analog is used to describe a waveform that's continuous in time and can take on a continuous range of amplitude values.

An example of an analog signal is some voltage that can be applied to an oscilloscope, resulting in a continuous display as a function of time. Analog signals can also be applied to a conventional spectrum analyzer to determine their frequency content. The term analog appears to have stemmed from the analog computers used prior to These computers solved linear differential equations by means of connecting physical electronic differentiators and integrators using old-style telephone operator patch cords.

That way, a continuous voltage or current in the actual circuit was analogous to some variable in a differential equation, such as speed, temperature, air pressure, etc. Although the flexibility and speed of modern-day digital computers have since made analog computers obsolete, a good description of the short-lived utility of analog computers can be found in reference .

Because present-day signal processing of continuous radio-type signals using resistors, capacitors, operational amplifiers, etc. The more correct term is continuous signal processing for what is today so commonly called analog signal processing. As such, in this book we'll minimize the use of the term analog signals and substitute the phrase continuous signals whenever appropriate. The term discrete-time signal is used to describe a signal whose independent time variable is quantized so that we know only the value of the signal at discrete instants in time.

Thus a discrete-time signal is not represented by a continuous waveform but, instead, a sequence of values. In addition to quantizing time, a discrete-time signal quantizes the signal amplitude.

We can illustrate this concept with an example. Think of a continuous sinewave with a peak amplitude of 1 at a frequency f o described by the equation. The frequency f o is measured in hertz Hz. In physical systems, we usually measure frequency in units of hertz.