# Understand the principles of operation and characterization of digital filters

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Digital Filters 1.0 Aim Understand the principles of operation and characterization of digital filters 2.0 Learning Outcomes You will be able to: Implement a digital filter in MATLAB. Investigate how commercial filters work Describe the transfer function of a filter A. Introduction Suppose the "raw" signal which is to be digitally filtered is in the form of a voltage waveform described by the function V = x (t) where t is time. This signal is sampled at time intervals h (the sampling interval). The sampled value at time t = ih is x i = x (ih) Thus the digital values transferred from the ADC to the processor can be represented by the sequence x 0, x 1, x 2, x 3,... corresponding to the values of the signal waveform at times t = 0, h, 2h, 3h,... (where t = 0 is the instant at which sampling begins). At time t = nh (where n is some positive integer), the values available to the processor, stored in memory, are x 0, x 1, x 2, x 3,..., x n Note that the sampled values xn+1, xn+2 etc. are not available as they haven't happened yet! The digital output from the processor to the DAC consists of the sequence of values y 0, y 1, y 2, y 3,..., y n In general, the value of yn is calculated from the values x 0, x 1, x 2, x 3,..., x n. The way in which the y's are calculated from the x's determines the filtering action of the digital filter.

2 Examples of simple digital filters The following examples illustrate the essential features of digital filters. 1. UNITY GAIN FILTER: y n = x n Each output value yn is exactly the same as the corresponding input value xn: y 0 = x 0 y 1 = x 1 y 2 = x 2 This is a trivial case in which the filter has no effect on the signal. 2. SIMPLE GAIN FILTER: y n = Kx n (K = constant) This simply applies a gain factor K to each input value: y 0 = Kx 0 y 1 = Kx 1 y 2 = Kx 2 K > 1 makes the filter an amplifier, while 0 < K < 1 makes it an attenuator. K < 0 corresponds to an inverting amplifier. Example (1) above is the special case where K = PURE DELAY FILTER: y n = x n-1 The output value at time t = nh is simply the input at time t = (n-1)h, i.e. the signal is delayed by time h: y 0 = x -1 y 1 = x 0 y 2 = x 1 y 3 = x 2 Note that as sampling is assumed to commence at t = 0, the input value x -1 at t = -h is undefined. It is usual to take this (and any other values of x prior to t = 0) as zero. 4. TWO-TERM DIFFERENCE FILTER: y n = x n - x n-1 The output value at t = nh is equal to the difference between the current input x n and the previous input x n-1 :

3 y 0 = x 0 - x -1 y 1 = x 1 - x 0 y 2 = x 2 - x 1 y 3 = x 3 - x 2 i.e. the output is the change in the input over the most recent sampling interval h. The effect of this filter is similar to that of an analog differentiator circuit. 5. TWO-TERM AVERAGE FILTER: y n = (x n + x n-1 ) / 2 The output is the average (arithmetic mean) of the current and previous input: y 0 = (x 0 + x -1 ) / 2 y 1 = (x 1 + x 0 ) / 2 y 2 = (x 2 + x 1 ) / 2 y 3 = (x 3 + x 2 ) / 2 This is a simple type of low pass filter as it tends to smooth out high-frequency variations in a signal. (We will look at more effective low pass filter designs later). 6. THREE-TERM AVERAGE FILTER: y n = (x n + xn-1 + x n-2 ) / 3 This is similar to the previous example, with the average being taken of the current and two previous inputs: y 0 = (x 0 + x -1 + x -2 ) / 3 y 1 = (x 1 + x 0 + x -1 ) / 3 y 2 = (x 2 + x 1 + x 0 ) / 3 y 3 = (x 3 + x 2 + x 1 ) / 3 As before, x-1 and x-2 are taken to be zero. 7. CENTRAL DIFFERENCE FILTER: y n = (x n - x n-2 ) / 2 This is similar in its effect to example (4). The output is equal to half the change in the input signal over the previous two sampling intervals: y 0 = (x 0 - x -2 ) / 2 y 1 = (x 1 - x -1 ) / 2 y 2 = (x 2 - x 0 ) / 2 y 3 = (x 3 - x 1 ) / 2

4 Order of a digital filter The order of a digital filter can be defined as the number of previous inputs (stored in the processor's memory) used to calculate the current output. This is illustrated by the filters given as examples in the previous section. Examples (1-2): These are zero order filters, since the current output depends only on the current input and not on any previous inputs. Example (3-5): These are first order filters, as one previous input (n-1) is required to calculate current sample. (Note that in (3) the filter is classed as first-order because it uses one previous input, even though the current input is not used). Example (6-7): To compute the current output, two previous inputs are needed; this is therefore a second-order filter. The order of a digital filter may be any positive integer. A zero-order filter is possible, but somewhat trivial, since it does not really filter the input signal in the accepted sense. Digital filter coefficients All of the digital filter examples given in the previous section can be written in the following general forms: y n = a 0 x n + a 1 x n-1 + a 2 x n-2 Similar expressions can be developed for filters of any order. The constants a 0, a 1, a 2,... appearing in these expressions are called the filter coefficients. The values of these coefficients determine the characteristics of a particular filter. Recursive and non-recursive filters For all the examples of digital filters discussed so far, the current output is calculated solely from the current and previous input values. This type of filter is said to be nonrecursive. A recursive filter is one that in addition to input values also uses previous output values. These, like the previous input values, are stored in the processor's memory. The word recursive literally means "running back", and refers to the fact that previouslycalculated output values go back into the calculation of the latest output. The expression

5 for a recursive filter therefore contains not only terms involving the input values (xn, xn- 1, xn-2,...) but also terms in yn-1, yn-2,... Example of a recursive filter A simple example of a recursive digital filter is given by y n = x n + y n-1 In other words, this filter determines the current output by adding the current input (xn) to the previous output. Thus: y0 = x0 + y-1 y1 = x1 + y0 y2 = x2 + y1 y3 = x3 + y2 Note that y -1 (like x -1 ) is undefined, and is usually taken to be zero. Let us consider the effect of this filter in more detail. If in each of the above expressions we substitute for yn-1 the value given by the previous expression, we get the following: y 0 = x 0 + y -1 = x 0 y 1 = x 1 + y 0 = x 1 + x 0 y 2 = x 2 + y 1 = x 2 + x 1 + x 0 y 3 = x 3 + y 2 = x 3 + x 2 + x 1 + x 0 Thus we can see that yn, the output at t = nh, is equal to the sum of the current input x n and all the previous inputs. This filter therefore sums or integrates the input values, and so has a similar effect to an analog integrator circuit. This example demonstrates an important and useful feature of recursive filters: the economy with which the output values are calculated, as compared with the equivalent non-recursive filter. In this example, each output is determined simply by adding two numbers together. FIR and IIR filters Some people prefer an alternative terminology in which a non-recursive filter is known as an FIR (or Finite Impulse Response) filter, and a recursive filter as an IIR (or Infinite

6 Impulse Response) filter. These terms refer to the differing "impulse responses" of the two types of filter The impulse response of a digital filter is the output sequence from the filter when a unit impulse is applied at its input. (A unit impulse is a very simple input sequence consisting of a single value of 1 at time t = 0, followed by zeros at all subsequent sampling instants). An FIR filter is one whose impulse response is of finite duration. An IIR filter is one whose impulse response (theoretically) continues for ever, because the recursive (previous output) terms feed back energy into the filter input and keep it going. The term IIR is not very accurate, because the actual impulse responses of nearly all IIR filters reduce virtually to zero in a finite time. Nevertheless, these two terms are widely used. Order of a recursive (IIR) digital filter The order of a digital filter was defined earlier as the number of previous inputs that have to be stored in order to generate a given output. This definition is appropriate for nonrecursive (FIR) filters, which use only the current and previous inputs to compute the current output. In the case of recursive filters, the definition can be extended as follows: The order of a recursive filter is the largest number of previous input or output values required to compute the current output. For example, the recursive filter discussed above, given by the expression y n = x n + y n-1 is classed as being of first order. Coefficients of recursive (IIR) digital filters From the above discussion, we can see that a recursive filter is basically like a nonrecursive filter, with the addition of extra terms involving previous outputs (y n-1, y n-2 etc.). The general form of recursive filter with M recursive (output) and N direct (input) elements is b 0 y n + b 1 y n b M y n-m = a 0 x n + a 1 x n-1 + a N x n-n Note the convention that the coefficients of the inputs (the x's) are denoted by a's, while the coefficients of the outputs (the y's) are denoted by b's. The order of IIR filter is max(n,m)

7 Implementation of Digital Filters Output from a digital filter is made up from previous inputs and previous outputs. This operation can be written as weighted moving average between filter coefficients and a signal flipped back to front. This summation is called the operation of convolution The filter can also be drawn as a following block diagram The filter diagram can show what hardware elements will be required when implementing the filter:

8 Commercial DSP chips are specifically designed to perform fast buffering, additions and multiplications of multiple data in one clock cycle. The precision of the output depends on the precision of the numerical representation of the coefficients, the samples and the operation results. Finite precision of these elements leads to quantization errors. So apart from errors when measuring signals, the following errors arise within the hardware used for processing: errors in arithmetic within the hardware (for example 16 bit fixed point roundoff) trncation when results are stored (most DSP processors have extended registers internally, so truncation usually occurs when results are stored to memory) quantisation of filter coefficients which have to be stored in memory B. The transfer function of a digital filter In the last section, we used two different ways of expressing the action of a digital filter: a form giving the output yn directly, and a "symmetrical" form with all the output terms (y's) on one side and all the input terms (x's) on the other. In this section, we introduce what is called the transfer function of a digital filter. This is obtained from the symmetrical form of the filter expression, and it allows us to describe a filter by means of a convenient, compact expression. The transfer function of a filter can be used to determine many of the characteristics of the filter, such as its frequency response. The unit delay operator First of all, we must introduce the unit delay operator, denoted by the symbol z -1 When applied to a sequence of digital values, this operator gives the previous value in the sequence. It therefore in effect introduces a delay of one sampling interval. Applying the operator z -1 to an input value (say x n ) gives the previous input (x n-1 ): z -1 x n = x n-1 Suppose we have an input sequence x 0 = 5 x 1 = -2 x 2 = 0 x 3 = 7 x 4 = 10

9 Then z -1 x 1 = 5 z -1 x 2 = -2 z -1 x 3 = 0 and so on. Note that z -1 x 0 would be x -1 which is unknown (and usually taken to be zero, as we have already seen). Similarly, applying the z -1 operator to an output gives the previous output: z -1 y n = y n-1 Applying the delay operator z -1 twice produces a delay of two sampling intervals. We adopt the convention (soon to be explained why) z -1 z -1 = z -2 i.e. the operator z -2 represents a delay of two sampling intervals: z -2 x n = x n-2 This notation can be extended to delays of three or more sampling intervals, the appropriate power of z -1 being used. Let us now use this notation in the description of a recursive digital filter. Consider, for example, a general second-order filter expressed using the delay operator given in its symmetrical form (b 0 + b 1 z -1 + b 2 z -2 ) y n = (a 0 + a 1 z -1 + a 2 z -2 ) x n Rearranging this to give a direct relationship between the output and input for the filter, we get y n / x n = (a 0 + a 1 z -1 + a 2 z -2 ) / (b0 + b 1 z -1 + b 2 z -2 ) This is the general form of the transfer function for a second-order recursive (IIR) filter. Transfer function examples 1. The three-term average filter, defined by the expression y n = 1/3 (x n + x n-1 + x n-2 ) can be written using the z-1 operator notation as

10 y n = 1/3 (x n + z -1 x n + z -2 x n ) = 1/3 (1 + z -1 + z -2 ) x n The transfer function for the filter is therefore y n / x n = 1/3 (1 + z -1 + z -2 ) 2. The general form of the transfer function for a first-order recursive filter can be written y n / x n = (a 0 + a 1 z -1 ) / (b 0 + b 1 z -1 ) Exercise: Write done transfer function of a simple first-order recursive filter y n = x n + y n-1 3. In MATLAB it is common to express the coefficients as vectors A and B. As a further example, consider the second-order IIR filter with coefficients B = [1 2-1], A=[1 2 1]. Expressing this filter in terms of the delay operator gives (1 + 2z -1 - z -2 ) y n = (1 + 2z -1 + z -2 ) x n and so the transfer function is y n / x n = (1 + 2z -1 + z -2 ) / (1 + 2z -1 - z -2 ) Delaying a phasor Filters operate by combining delayed version of input and output signals. The operation of a delay line is best understood using a complex periodic signal called phasor, expressed in terms of continuous time parameter t (instead of index n) e j"t This is a rotating vector on the unit circle (complex number of magnitue 1) that completes one full cycle in T = 2" /# seconds. Note that we express here the imaginary unit number by j = "1. (It is commonly expressed also using the letter i, so we might use i or j interchangeably) If we delay the phase by " seconds, we get

11 e j"(t#\$ ) = e # j"\$ e j"t Note that" in the exponent represents a fixed delay time. It should not be confused with the time parameter t in the exponent of the phasor that keeps increasing (time goes on). A delay then becomes a multiplication by a constant complex number e j"#. In case of a single sample delay, the multiplicative delay factor is written using the symbol z z = e " j# Example: For a FIR order one filter y t = a o x t + a 1 x t"1 the effect of applying a delay to input that is a phasor x t = e " j#t becomes y t = [a 0 + a 1 e " j# ]x t The multiplicative expression H(z) = a 0 + a 1 e " j# is called the filter Transfer Function. Describing filters in terms of Transfer Function. Filters are made of weighted sums of delayed input and output samples. When filter is applied to a general signal, we used the unit delay notation to represent the filter in an elegant way as weighted sums of repeated unit delay operations. So far the unit delay appeared as a new mathematical operation that alters the time index n of input or output samples (x(n) and y(n)). As we just saw, in the special case of a phasor signal the delay actually becomes a multiplication by a complex number that has a unit magnitude and whose angle is equal to the amount of time delay that is being applied to that phasor.

12 What is interesting and important in understanding the effect of a filter on a phasor is that the now the operation of any filter on any signal can be described in terms of transfer function. How? Using transfer function, the effect of the filter on a phasor is written mathematically in terms of multiplication of the phasor by complex polynomial or by a ratio of complex polynomials. The coefficients of this polynomial are equal to coefficients of the filter, with forward filter coefficients appearing at the numerator and backward (recursive) filter coefficients at the denominator, and the delays are the negative powers of z. When considering some other signal, it still can be described in terms of a combination of complex phasors. This is done by writing any signal in terms of a weighted sum of phasors and different frequency. The coefficients (amplitudes and phases) of these phasors are found by transforming the signal into a complex domain using the Fourier Transform. In the frequency domain, the delay operation becomes multiplication of each phasor component by a transfer function. This is also called the system Z-transform. In summary: Using Fourier Transform allows any signal in terms of combination of phasor. This allows expressing the output of a filter in terms of a multiplication between signal input frequency elements and the filter transfer function. The filtering operation can be described in the frequency domain as weighted combination of delays of the phasor components of the input x(n) signals. We will learn about Fourier Transforms in the next section. The z-plane Once the Z-transform of a system has been determined, one can use the information contained in function's polynomials to graphically represent the function and easily observe many defining characteristics. H(z) = P(z) Q(z) Zeros 1. The value(s) for z where P(z)=0 2. The complex frequencies that make the overall gain of the filter transfer function zero. Poles 1. The value(s) for z where Q(z)=0 2. The complex frequencies that make the overall gain of the filter transfer function infinite. Example 1

13 This is a simple transfer function with the poles and zeros shown below it. Hz=(z+1)/[(z 1/2)(z+3/4)] The zeros are: {-1}, The poles are: {1/2, -3/4} Once the poles and zeros have been found for a given Z-Transform, they can be plotted onto the Z-Plane. The Z-plane is a complex plane with an imaginary and real axis referring to the complex-valued variable z. The position on the complex plane is given the radio and the angle from the positive, real axis around the plane. When mapping poles and zeros onto the plane, poles are denoted by an "x" and zeros by an "o". The below figure shows the Z-Plane, and examples of plotting zeros and poles onto the plane can be found in the following section. Examples of Pole/Zero Plots Plotting z-plan in MATLAB % Set up vector for zeros z = [j ; -j]; % Set up vector for poles p = [-1 ;.5+.5j ;.5-.5j]; figure(1); zplane(z,p); title('pole/zero Plot for Complex Pole/Zero Plot Example'); Frequency Response and the Z-Plane

14 The reason it is helpful to understand and create these pole/zero plots is due to their ability to help us easily design a filter. Based on the location of the poles and zeros, the magnitude response of the filter can be quickly understood. Also, by starting with the pole/zero plot, one can design a filter and obtain its transfer function very easily. Refer to this module for information on the relationship between the pole/zero plot and the frequency response. Estimating the Frequency Response of an IIR Filter in MATLAB %***Low Pass Averaging Filter*** % % y(n) = y(n-1) + 0.1[x(n) - y(n-1)] % %H(Z) = 0.1Z/(Z - 0.9) num = [0.1] den = [1-0.9] Fs = 10 freqz(num, den, 128, Fs) Additional reading: Steiglitz book 4.7, 4.8, 5.3, 5.4, 5.5, 5.7, 6.1

### What is a Filter? Output Signal. Input Signal Amplitude. Frequency. Low Pass Filter

What is a Filter? Input Signal Amplitude Output Signal Frequency Time Sequence Low Pass Filter Time Sequence What is a Filter Input Signal Amplitude Output Signal Frequency Signal Noise Signal Noise Frequency

### POLES, ZEROS, and HIGHER ORDER FILTERS By John Ehlers

POLES, ZEROS, and HIGHER ORDER FILTERS By John Ehlers INTRODUCTION Smoothing filters are at the very core of what we technical analysts do. All technicians are familiar with Simple Moving Averages, Exponential

### Chapter 4: Time-Domain Representation for Linear Time-Invariant Systems

Chapter 4: Time-Domain Representation for Linear Time-Invariant Systems In this chapter we will consider several methods for describing relationship between the input and output of linear time-invariant

### Module 4. Contents. Digital Filters - Implementation and Design. Signal Flow Graphs. Digital Filter Structures. FIR and IIR Filter Design Techniques

Module 4 Digital Filters - Implementation and Design Digital Signal Processing. Slide 4.1 Contents Signal Flow Graphs Basic filtering operations Digital Filter Structures Direct form FIR and IIR filters

### Department of Electronics and Communication Engineering 1

DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING III Year ECE / V Semester EC 6502 PRINCIPLES OF DIGITAL SIGNAL PROCESSING QUESTION BANK Department of

### EE 422G - Signals and Systems Laboratory

EE 4G - Signals and Systems Laboratory Lab 4 IIR Filters written by Kevin D. Donohue Department of Electrical and Computer Engineering University of Kentucky Lexington, KY 40506 September 6, 05 Objectives:

### Chapter 4: Problem Solutions

Chapter 4: Problem s Digital Filters Problems on Non Ideal Filters à Problem 4. We want to design a Discrete Time Low Pass Filter for a voice signal. The specifications are: Passband F p 4kHz, with 0.8dB

### Difference Equations and Digital Filters

SIGNALS AND SYSTEMS: PAPER C HANDOUT 8. Dr David Corrigan. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.mee.tcd.ie/ corrigad Difference Equations and Digital Filters The last topic discussed

### Digital Filter Design (FIR) Using Frequency Sampling Method

Digital Filter Design (FIR) Using Frequency Sampling Method Amer Ali Ammar Dr. Mohamed. K. Julboub Dr.Ahmed. A. Elmghairbi Electric and Electronic Engineering Department, Faculty of Engineering Zawia University

### Design of Efficient Digital Interpolation Filters for Integer Upsampling. Daniel B. Turek

Design of Efficient Digital Interpolation Filters for Integer Upsampling by Daniel B. Turek Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements

### Em bedded DSP : I ntroduction to Digital Filters

Embedded DSP : Introduction to Digital Filters 1 Em bedded DSP : I ntroduction to Digital Filters Digital filters are a important part of DSP. In fact their extraordinary performance is one of the keys

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur-603 203 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6502 PRINCIPAL OF DIGITAL SIGNAL PROCESSING YEAR / SEMESTER: III / V ACADEMIC

### Outline. Digital Control System Models. ADC Model. Common Digital Control System Configuration

Outline Digital Control System Models M. Sami Fadali Professor of Electrical Engineering University of Nevada Model of ADC. Model of DAC. Model of ADC, analog subsystem and DAC. Systems with transport

### FIR Filter Design. FIR Filters and the z-domain. The z-domain model of a general FIR filter is shown in Figure 1. Figure 1

FIR Filters and the -Domain FIR Filter Design The -domain model of a general FIR filter is shown in Figure. Figure Each - box indicates a further delay of one sampling period. For example, the input to

### Lecture 06: Design of Recursive Digital Filters

Lecture 06: Design of Recursive Digital Filters John Chiverton School of Information Technology Mae Fah Luang University 1st Semester 2009/ 2552 Lecture Contents Introduction IIR Filter Design Pole-Zero

### IIR Filter. Dmitry Teytelman and Dan Van Winkle. Student name: RF and Digital Signal Processing US Particle Accelerator School June, 2009

Dmitry Teytelman and Dan Van Winkle Student name: RF and Digital Signal Processing US Particle Accelerator School 15 19 June, 2009 June 17, 2009 Contents 1 Introduction 2 2 IIR Coefficients 2 3 Exercises

### LAB 4 DIGITAL FILTERING

LAB 4 DIGITAL FILTERING 1. LAB OBJECTIVE The purpose of this lab is to design and implement digital filters. You will design and implement a first order IIR (Infinite Impulse Response) filter and a second

### A Brief Introduction to Sigma Delta Conversion

A Brief Introduction to Sigma Delta Conversion Application Note May 995 AN954 Author: David Jarman Introduction The sigma delta conversion techniue has been in existence for many years, but recent technological

### Recursive Filters. The Recursive Method

CHAPTER 19 Recursive Filters Recursive filters are an efficient way of achieving a long impulse response, without having to perform a long convolution. They execute very rapidly, but have less performance

### Frequency Response of FIR Filters

Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

### Sampling and Reconstruction of Analog Signals

Sampling and Reconstruction of Analog Signals Chapter Intended Learning Outcomes: (i) Ability to convert an analog signal to a discrete-time sequence via sampling (ii) Ability to construct an analog signal

### Department of Electrical and Computer Engineering Ben-Gurion University of the Negev. LAB 1 - Introduction to USRP

Department of Electrical and Computer Engineering Ben-Gurion University of the Negev LAB 1 - Introduction to USRP - 1-1 Introduction In this lab you will use software reconfigurable RF hardware from National

### The Discrete Fourier Transform

The Discrete Fourier Transform Introduction The discrete Fourier transform (DFT) is a fundamental transform in digital signal processing, with applications in frequency analysis, fast convolution, image

### Recursive Gaussian filters

CWP-546 Recursive Gaussian filters Dave Hale Center for Wave Phenomena, Colorado School of Mines, Golden CO 80401, USA ABSTRACT Gaussian or Gaussian derivative filtering is in several ways optimal for

### ECG-Amplifier. MB Jass 2009 Daniel Paulus / Thomas Meier. Operation amplifier (op-amp)

ECG-Amplifier MB Jass 2009 Daniel Paulus / Thomas Meier Operation amplifier (op-amp) Properties DC-coupled High gain electronic ec c voltage amplifier Inverting / non-inverting input and single output

### ENEE 425 Spring 2010 Assigned Homework Oppenheim and Shafer (3rd ed.) Instructor: S.A. Tretter

ENEE 425 Spring 2010 Assigned Homework Oppenheim and Shafer (3rd ed.) Instructor: S.A. Tretter Note: The dates shown are when the problems were assigned. Homework will be discussed in class at the beginning

### DIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION

DIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION Introduction The outputs from sensors and communications receivers are analogue signals that have continuously varying amplitudes. In many systems

### Frequency response of a general purpose single-sided OpAmp amplifier

Frequency response of a general purpose single-sided OpAmp amplifier One configuration for a general purpose amplifier using an operational amplifier is the following. The circuit is characterized by:

### CONVERTERS. Filters Introduction to Digitization Digital-to-Analog Converters Analog-to-Digital Converters

CONVERTERS Filters Introduction to Digitization Digital-to-Analog Converters Analog-to-Digital Converters Filters Filters are used to remove unwanted bandwidths from a signal Filter classification according

### 1 1. The ROC of the z-transform H(z) of the impulse response sequence h[n] is defined

A causal LTI digital filter is BIBO stable if and only if its impulse response h[ is absolutely summable, i.e., S h [ < n We now develop a stability condition in terms of the pole locations of the transfer

### Moving Average Filters

CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average

### (Refer Slide Time: 01:11-01:27)

Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 6 Digital systems (contd.); inverse systems, stability, FIR and IIR,

### This is the 23 rd lecture on DSP and our topic for today and a few more lectures to come will be analog filter design. (Refer Slide Time: 01:13-01:14)

Digital Signal Processing Prof. S.C. Dutta Roy Department of Electrical Electronics Indian Institute of Technology, Delhi Lecture - 23 Analog Filter Design This is the 23 rd lecture on DSP and our topic

### Digital Data Transmission Notes

1 Digital Communication Systems The input to a digital communication system is a sequence of digits. The input could be sourced by a data set, computer, or quantized and digitized analog signal. Such digital

### Chapter 3 Discrete-Time Fourier Series. by the French mathematician Jean Baptiste Joseph Fourier in the early 1800 s. The

Chapter 3 Discrete-Time Fourier Series 3.1 Introduction The Fourier series and Fourier transforms are mathematical correlations between the time and frequency domains. They are the result of the heat-transfer

### Design Technique of Bandpass FIR filter using Various Window Function

IOSR Journal of Electronics and Communication Engineering (IOSR-JECE) e-issn: 2278-2834,p- ISSN: 2278-8735. Volume 6, Issue 6 (Jul. - Aug. 213), PP 52-57 Design Technique of Bandpass FIR filter using Various

### DSP Laboratory Work S. Laboratory exercises with TMS320C5510 DSK

DSP Laboratory Work 521485S Laboratory exercises with TMS320C5510 DSK Jari Hannuksela Information Processing Laboratory Dept. of Electrical and Information Engineering, University of Oulu ovember 14, 2008

### Why z-transform? Convolution becomes a multiplication of polynomials

z-transform Why z-transform? The z-transform introduces polynomials and rational functions in the analysis of linear time-discrete systems and has a similar importance as the Laplace transform for continuous

### The Audio Engineer s Approach to Understanding Digital Filters For the idiot such as myself

The Audio Engineer s Approach to Understanding Digital Filters For the idiot such as myself By Nika Aldrich The following is a paper written for the benefit of the audio engineer who yearns to understand

### Introduction p. 1 Digital signal processing and its benefits p. 1 Application areas p. 3 Key DSP operations p. 5 Digital signal processors p.

Preface p. xv Introduction p. 1 Digital signal processing and its benefits p. 1 Application areas p. 3 Key DSP operations p. 5 Digital signal processors p. 13 Overview of real-world applications of DSP

### chapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective

Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything

### The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #1. Spirou et Fantasio

The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #1 Date: October 14, 2016 Course: EE 445S Evans Name: Spirou et Fantasio Last, First The exam is scheduled to last

### Roundoff Noise in IIR Digital Filters

Chapter 16 Roundoff Noise in IIR Digital Filters It will not be possible in this brief chapter to discuss all forms of IIR (infinite impulse response) digital filters and how quantization takes place in

### 2 Background: Fourier Series Analysis and Synthesis

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Spring 2001 Lab #11: Design with Fourier Series Date: 3 6 April 2001 This is the official Lab #11 description. The

### This is the 31 st lecture and we continue our discussion on Lattice Synthesis. (Refer Slide Time: 01:13)

Digital Signal Processing Prof: S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture 31 Lattice Synthesis (Contd...) This is the 31 st lecture and we continue

### Digital Representations of Analog Systems for Control System Applications

Digital Representations of Analog Systems for Control System Applications Peter Way May 21, 29 This tutorial shows how digital controllers can be created from an analog (or continuous) model. This is especially

### Master Thesis of 30 Credits

LINNAEUS UNIVERSITY IFE, VÄXJÖ Department of Physics and Electrical Engineering Master Thesis of 30 Credits Finite Precision Error in FPGA Measurement Submitted By: Shoaib Ahmad Registration No: 850302-7578

### FILTER CIRCUITS. A filter is a circuit whose transfer function, that is the ratio of its output to its input, depends upon frequency.

FILTER CIRCUITS Introduction Circuits with a response that depends upon the frequency of the input voltage are known as filters. Filter circuits can be used to perform a number of important functions in

### Multiband Digital Filter Design for WCDMA Systems

2011 International Conference on Circuits, System and Simulation IPCSIT vol.7 (2011) (2011) IACSIT Press, Singapore Multiband Digital Filter Design for WCDMA Systems Raweewan Suklam and Chaiyod Pirak The

### The Z transform (3) 1

The Z transform (3) 1 Today Analysis of stability and causality of LTI systems in the Z domain The inverse Z Transform Section 3.3 (read class notes first) Examples 3.9, 3.11 Properties of the Z Transform

### IIR Filter design (cf. Shenoi, 2006)

IIR Filter design (cf. Shenoi, 2006) The transfer function of the IIR filter is given by Its frequency responses are (where w is the normalized frequency ranging in [ π, π]. When a and b are real, the

### Difference Equations

Difference Equations Andrew W H House 10 June 004 1 The Basics of Difference Equations Recall that in a previous section we saw that IIR systems cannot be evaluated using the convolution sum because it

### Filter Comparison. Match #1: Analog vs. Digital Filters

CHAPTER 21 Filter Comparison Decisions, decisions, decisions! With all these filters to choose from, how do you know which to use? This chapter is a head-to-head competition between filters; we'll select

### Discrete-time signals and systems

Chapter 2 Discrete-time signals and systems Contents Overview........................................................ 2.2 Discrete-time signals.................................................. 2.2 Some

### CHAPTER 6 Frequency Response, Bode Plots, and Resonance

ELECTRICAL CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. State the fundamental concepts of Fourier analysis. 2. Determine the output of a filter for a given input consisting of sinusoidal

### Lab 1 Filter Design and Evaluation in MATLAB

Lab 1 Filter Design and Evaluation in MATLAB S2E DSP Systems In Practice Andre Lundkvist nadlun-5@student.ltu.se Rikard Qvarnström rikqva-5@student.ltu.se Luleå, 6th October 29 Abstract This report is

### Understanding Poles and Zeros

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

### This is the 39th lecture and our topic for today is FIR Digital Filter Design by Windowing.

Digital Signal Processing Prof: S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 39 FIR Digital Filter Design by Windowing This is the 39th lecture and

### Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs

Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in

### Analysis and Design of FIR filters using Window Function in Matlab

International Journal of Computer Engineering and Information Technology VOL. 3, NO. 1, AUGUST 2015, 42 47 Available online at: www.ijceit.org E-ISSN 2412-8856 (Online) Analysis and Design of FIR filters

### The Fourier Analysis Tool in Microsoft Excel

The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier

### The Filter Wizard issue 28: Analyze FIR Filters Using High-School Algebra Kendall Castor-Perry

The Filter Wizard issue 28: Analyze FIR Filters Using High-School Algebra Kendall Castor-Perry In this first part of a two-parter, the Filter Wizard looks at some fundamental facts about FIR filter responses,

### Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes

Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes (i) Understanding the relationships between the transform, discrete-time Fourier transform (DTFT), discrete Fourier

### The z-transform. The Nature of the z-domain. X(s) '

CHAPTER 33 The z-transform Just as analog filters are designed using the Laplace transform, recursive digital filters are developed with a parallel technique called the z-transform. The overall strategy

### PROPERTIES AND EXAMPLES OF Z-TRANSFORMS

PROPERTIES AD EXAMPLES OF Z-TRASFORMS I. Introduction In the last lecture we reviewed the basic properties of the z-transform and the corresponding region of convergence. In this lecture we will cover

### OSE801 Engineering System Identification Fall 2012 Lecture 5: Fourier Analysis: Introduction

OSE801 Engineering System Identification Fall 2012 Lecture 5: Fourier Analysis: Introduction Instructors: K. C. Park and I. K. Oh (Division of Ocean Systems Engineering) System-Identified State Space Model

### (a) FIR Filter 6. (b) Feedback Filter 1.5. magnitude 4. magnitude omega/pi omega/pi

DSP First Laboratory Exercise #10 The z, n, and ^! Domains 1 Objective The objective for this lab is to build an intuitive understanding of the relationship between the location of poles and zeros in the

### Wearable ECG Patch. 1 Introduction. 2 ECG patch. Freescale Semiconductor, Inc. Application Note. Document Number: AN4963 Rev.

Freescale Semiconductor, Inc. Application Note Document Number: AN4963 Rev. 0, 08/2014 Wearable ECG Patch by José Santiago López Ramírez 1 Introduction Electronic products are becoming smaller and more

### IIR Filters. The General IIR Difference Equation. Chapter

Chapter IIR Filters 8 In this chapter we finally study the general infinite impulse response (IIR) difference equation that was mentioned back in Chapter 5. The filters will now include both feedback and

### SECTION 6 DIGITAL FILTERS

SECTION 6 DIGITAL FILTERS Finite Impulse Response (FIR) Filters Infinite Impulse Response (IIR) Filters Multirate Filters Adaptive Filters 6.a 6.b SECTION 6 DIGITAL FILTERS Walt Kester INTRODUCTION Digital

### Bharathwaj Muthuswamy EE100 Active Filters

Bharathwaj Muthuswamy EE100 mbharat@cory.eecs.berkeley.edu 1. Introduction Active Filters In this chapter, we will deal with active filter circuits. Why even bother with active filters? Answer: Audio.

### Today. Data acquisition Digital filters and signal processing. Filter examples and properties FIR filters Filter design. DACs PWM

Today Data acquisition Digital filters and signal processing Filter examples and properties FIR filters Filter design Implementation issues DACs PWM Data Acquisition Systems Many embedded systems measure

### Outline. FIR Filter Characteristics Linear Phase Windowing Method Frequency Sampling Method Equiripple Optimal Method Design Examples

FIR Filter Design Outline FIR Filter Characteristics Linear Phase Windowing Method Frequency Sampling Method Equiripple Optimal Method Design Examples FIR Filter Characteristics FIR difference equation

### Ch. 10 Infinite Impulse Response (IIR) Digital Filters. Outline

Ch. 10 Infinite Impulse Response (IIR) Digital Filters 1 Introduction IIR Theory IIR Coefficient Computation IIR Filter Implementation Fast IIR Filter Outline 2 1 Introduction The most important properties

### Polynomials and the Fast Fourier Transform (FFT) Battle Plan

Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and point-value representation

### HYBRID FIR-IIR FILTERS By John F. Ehlers

HYBRID FIR-IIR FILTERS By John F. Ehlers Many traders have come to me, asking me to make their indicators act just one day sooner. They are convinced that this is just the edge they need to make a zillion

### 2.161 Signal Processing: Continuous and Discrete Fall 2008

MT OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 00 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### 6.025J Medical Device Design Lecture 3: Analog-to-Digital Conversion Prof. Joel L. Dawson

Let s go back briefly to lecture 1, and look at where ADC s and DAC s fit into our overall picture. I m going in a little extra detail now since this is our eighth lecture on electronics and we are more

### Recognized simulation

From April 2005 High Frequency Electronics Copyright 2005 Summit Technical Media RTDX -Based Simulation Tools Support Development of Software-Defined Radio By Robert G. Davenport MC 2 Technology Group,

### Contents. I. Abstract 3. II. Introduction 4. III. Problem Statement 5. IV. Analysis. a. Developing the Appropriate Filter Structure 6

!"# \$%& Contents I. Abstract 3 II. Introduction 4 III. Problem Statement 5 IV. Analysis a. Developing the Appropriate Filter Structure 6 b. General Second Order Section Direct Form II Filter Analysis 7

### CM0340 SOLNS. Do not turn this page over until instructed to do so by the Senior Invigilator.

CARDIFF UNIVERSITY EXAMINATION PAPER Academic Year: 2008/2009 Examination Period: Examination Paper Number: Examination Paper Title: SOLUTIONS Duration: Autumn CM0340 SOLNS Multimedia 2 hours Do not turn

### Lab 4 Sampling, Aliasing, FIR Filtering

47 Lab 4 Sampling, Aliasing, FIR Filtering This is a software lab. In your report, please include all Matlab code, numerical results, plots, and your explanations of the theoretical questions. The due

### This representation is compared to a binary representation of a number with N bits.

Chapter 11 Analog-Digital Conversion One of the common functions that are performed on signals is to convert the voltage into a digital representation. The converse function, digital-analog is also common.

### where T o defines the period of the signal. The period is related to the signal frequency according to

SIGNAL SPECTRA AND EMC One of the critical aspects of sound EMC analysis and design is a basic understanding of signal spectra. A knowledge of the approximate spectral content of common signal types provides

### Approximation Errors in Computer Arithmetic (Chapters 3 and 4)

Approximation Errors in Computer Arithmetic (Chapters 3 and 4) Outline: Positional notation binary representation of numbers Computer representation of integers Floating point representation IEEE standard

### The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

### Analog to Digital, A/D, Digital to Analog, D/A Converters. An electronic circuit to convert the analog voltage to a digital computer value

Analog to Digital, A/D, Digital to Analog, D/A Converters An electronic circuit to convert the analog voltage to a digital computer value Best understood by understanding Digital to Analog first. A fundamental

### University of Rhode Island Department of Electrical and Computer Engineering ELE 436: Communication Systems. FFT Tutorial

University of Rhode Island Department of Electrical and Computer Engineering ELE 436: Communication Systems FFT Tutorial 1 Getting to Know the FFT What is the FFT? FFT = Fast Fourier Transform. The FFT

### FIR and IIR Transfer Functions

FIR and IIR Transfer Functions the input output relation of an LTI system is: the output in the z domain is: yn [ ] = hkxn [ ] [ k] k = Y( z) = H( z). X( z) Where H( z) = h[ n] z n= n so we can write the

### Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science : Discrete-Time Signal Processing

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture 8 DT Filter Design: IIR Filters Reading:

### Motorola Digital Signal Processors

Motorola Digital Signal Processors Principles of Sigma-Delta Modulation for Analog-to- Digital Converters by Sangil Park, Ph. D. Strategic Applications Digital Signal Processor Operation MOTOROLA APR8

### An Introduction to Modular Arbitrary Function Generators

An Introduction to Modular Arbitrary Function Generators Electronic test and measurements equipment can be classified into two major categories; measurement instruments and signal sources. Instruments

### Real-Time Filtering in BioExplorer

Real-Time Filtering in BioExplorer Filtering theory Every signal can be thought of as built up from a large number of sine and cosine waves with different frequencies. A digital filter is a digital signal

International Journal of Electronic and Electrical Engineering. ISSN 0974-2174, Volume 7, Number 4 (2014), pp. 379-386 International Research Publication House http://www.irphouse.com Hardware Implementation

### Aparna Tiwari 1, Vandana Thakre 2, Karuna Markam 3 *(Dept. of ECE, MITS, Gwalior,M.P.)

International Journal of Computer & Communication Engineering Research (IJCCER) Volume 2 - Issue 2 March 2014 Design Technique of Bandpass FIR filter using Various Function Aparna Tiwari 1, Vandana Thakre

### Chapter 21 Band-Pass Filters and Resonance

Chapter 21 Band-Pass Filters and Resonance In Chapter 20, we discussed low-pass and high-pass filters. The simplest such filters use RC components resistors and capacitors. It is also possible to use resistors

### (Refer Slide Time 6:09)

Digital Circuits and Systems Prof. S. Srinivasan Department of Electrical Engineering Indian Institute of Technology, Madras Lecture 3 Combinational Logic Basics We defined digital signal as a signal which