DOCSLIB.ORG

Lesson 7: Anti-Aliasing Filtering

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lesson 7: Anti-Aliasing Filtering 4/25/2016 Lesson 7: Anti- Aliasing Filtering ET 438b Sequential Control and Data Acquisition Department of Technology Lesson 7_et438b.pptx 1 Learning Objectives After this presentation you will be able to: Explain why anti-aliasing filters are used. Design a first order anti-aliasing filter using an OP AMP Design a 2nd order anti-aliasing filter using an OP AMP Verify the performance of an anti-aliasing filter using simulation software. Lesson 7_et438b.pptx 2 1 4/25/2016 Anti-Aliasing Filters Dealing with Aliasing in practical systems Exact frequency components of a sampled signal are unknown. Can not determine if signal component is alias or real Sampling rate limited by hardware selection. Lab DAQ cards rate is 200 kHz-250 kHz. fnyquist = 100 kHz – 125 kHz sets input frequency limits. Bandwidth limit input signals using anti-aliasing filters to eliminate frequencies above fnyquist. Lesson 7_et438b.pptx 3 Anti-Aliasing Filters Goal- reduce amplitude of all frequencies above fnyquist to zero level Ideal low pass filter Bode plot Cut-off Frequency, fc Set fc = fnyquist for perfect Amplitude Stop band signal elimination Frequency Pass band Cut-off Frequency, fc Practical (Butterworth) filters have sloping characteristics Amplitude 1st Order: -20 dB/decade nd 2 Order: -40 dB /decade Frequency 3rd Order: -60 db/decade Lesson 7_et438b.pptx 4 2 4/25/2016 Anti-Aliasing Filters: OP AMP Circuits First order Butterworth filter R2 Design formulas C1 R2 Av Voltage gain in the pass band -12V R1 U1 R1 UA741 1 Vin Vo1 fc Cut - off frequency (Hz) + 2R2C1 To DAQ Channel 12V Design Procedure 1.) Determine the acceptable level of signal gain, Av, at sampling frequency fs 2.) Use the formula below to determine the value of fc based of the designed signal gain f f s for A 1 c 2 v 1 Av 2 2 Av 3. Select a value of C1 from standard values and compute value of R2 Lesson 7_et438b.pptx 5 Anti-Aliasing Filters: OP AMP Circuits Design Procedure (continued) 4.) Set R1=R2 to give pass band gain of -1 (o dB). Amplify signal after filtering to reduce noise and unwanted signal components. (e.g. 60 Hz) Design Example: A data acquisition systems samples an analog signal at st Ts=0.0004 seconds. Design a 1 order anti-aliasing filter that will reduce the voltage level of all signal above the Nyquist frequency to 0.4 Solution: Determine the sampling frequency from Ts then find the fnyquist 1 1 fs 2500 Hz fs 2500 Hz fnyquist 1250 Hz Ts 0.0004 s 2 2 f 2500 Hz 2500 Hz f s 546 Hz Find the value of fc for the c 2 2 1 Av 1 0.4 2 5.25 given level of Av above 2 2 2 2 Av 0.4 fnyquist Lesson 7_et438b.pptx 6 3 4/25/2016 Design Example (continued) Select a capacitor value of 0.1 mF. Use larger values for lower f’s to keep 2700 values of resistors in range of 1k to 820k 0.1 mF R2 1 f 1 c 2R2C1 R2 C1 2C1fc 2700 1 -12V R2 2916 U1 6 R1 UA741 20.110 546 Vin Vo1 Set R1=R2 to give gain+ of -1 Select standard value of 2700 ohms and To DAQ Channel 12V Design complete. Check result with compute value of fc 1 circuit simulation. C1 590 Hz 20.1106 2700 @ 1.254 kHz Av=-7.4 dB (dB/20) dB=20Log(Av) 10 =Av (-7.4/20) 10 =Av=0.43 Lesson 7_et438b.pptx 7 Anti-Aliasing Filters- 2nd Order Filters Second order Butterworth filter C3 Unity gain is fixed by negative feedback loop in this design. 12V U2 R4 R5 UA741 + Vin Vo2 Use same design procedure as To DAQ Channel C2 -12V 1st order filter but use following formula to find cut-off frequency. f Design formulas f s for A 1 c 2 v 1 Av 4 2 2 Av 1 Voltage gain in the pass band Av 1 f Cut - off frequency (Hz) c 2 R4R5C3C2 2nd order filter produces -40 With R4 R5 and C3 2C2 dB/decade rolloff in stop band Lesson 7_et438b.pptx 8 4 4/25/2016 Anti-Aliasing 2nd Order Filter Design Design Example 2: Repeat the design of the 1st order filter example using a 2nd order filter nd Ts=0.0004 seconds sampling. Design a 2 order anti-aliasing filter that will reduce the voltage level of all signal above the Nyquist frequency to 0.4 Same fs and fnyquist as previous case f 2500 Hz 2500 Hz Find the value of fc for the f s 826 Hz nd c 2 2 4 given level of A using 2 1 Av 1 0.4 2 5.25 v 4 4 2 2 2 2 order filter formula Av 0.4 Lesson 7_et438b.pptx 9 Design Example (continued) Select value of C2 and compute C3. Let C2=0.1 mF so….. 0.2 mF C3 1.5 k C3= 2∙C2=2∙(0.1 mF)= 0.2 mF 1.5 k 12V U2 R4 R5 UA741 m + Vin 0.1 F Vo2 To DAQ Channel 1 C2 -12V R4 Use design formulas to find value of R4=R5 2 2C2fc 1 1 R4 6 fc With R4 R5 and C3 2C2 2 20.110 F826 Hz 2 R4R5C3C2 1 1 f 1 1 fs 2500 Hz R4 1363 R5 c f 2 2 2500 Hz f 1250 Hz 2 Rs 4 2C2 2 2 R4C2 nyquist Ts 0.0004 s 2 2 Use standard value of 1.5k. Solve for R4 given C2 and f Response very sensitive to R c values 1 1 R4 fc 2 2 R4C2 2 2C2fc Lesson 7_et438b.pptx 10 5 4/25/2016 Design Example (continued) Design complete. Check result with circuit simulation. @ 1.249 kHz Av=-9.4 dB (dB/20) dB=20Log(Av) 10 =Av (-9.4/20) 10 =Av=0.34 Flatter response in pass band sharper roll off. Lesson 7_et438b.pptx 11 End Lesson 7: Anti- Aliasing Filtering ET 438b Sequential Control and Data Acquisition Department of Technology Lesson 7_et438b.pptx 12 6 .
Load more

Recommended publications
  • Mathematical Basics of Bandlimited Sampling and Aliasing
    Signal Processing Mathematical basics of bandlimited sampling and aliasing Modern applications often require that we sample analog signals, convert them to digital form, perform operations on them, and reconstruct them as analog signals. The important question is how to sample and reconstruct an analog signal while preserving the full information of the original. By Vladimir Vitchev o begin, we are concerned exclusively The limits of integration for Equation 5 T about bandlimited signals. The reasons are specified only for one period. That isn’t a are both mathematical and physical, as we problem when dealing with the delta func- discuss later. A signal is said to be band- Furthermore, note that the asterisk in Equa- tion, but to give rigor to the above expres- limited if the amplitude of its spectrum goes tion 3 denotes convolution, not multiplica- sions, note that substitutions can be made: to zero for all frequencies beyond some thresh- tion. Since we know the spectrum of the The integral can be replaced with a Fourier old called the cutoff frequency. For one such original signal G(f), we need find only the integral from minus infinity to infinity, and signal (g(t) in Figure 1), the spectrum is zero Fourier transform of the train of impulses. To the periodic train of delta functions can be for frequencies above a. In that case, the do so we recognize that the train of impulses replaced with a single delta function, which is value a is also the bandwidth (BW) for this is a periodic function and can, therefore, be the basis for the periodic signal.
    [Show full text]
  • Efficient Supersampling Antialiasing for High-Performance Architectures
    Efficient Supersampling Antialiasing for High-Performance Architectures TR91-023 April, 1991 Steven Molnar The University of North Carolina at Chapel Hill Department of Computer Science CB#3175, Sitterson Hall Chapel Hill, NC 27599-3175 This work was supported by DARPA/ISTO Order No. 6090, NSF Grant No. DCI- 8601152 and IBM. UNC is an Equa.l Opportunity/Affirmative Action Institution. EFFICIENT SUPERSAMPLING ANTIALIASING FOR HIGH­ PERFORMANCE ARCHITECTURES Steven Molnar Department of Computer Science University of North Carolina Chapel Hill, NC 27599-3175 Abstract Techniques are presented for increasing the efficiency of supersampling antialiasing in high-performance graphics architectures. The traditional approach is to sample each pixel with multiple, regularly spaced or jittered samples, and to blend the sample values into a final value using a weighted average [FUCH85][DEER88][MAMM89][HAEB90]. This paper describes a new type of antialiasing kernel that is optimized for the constraints of hardware systems and produces higher quality images with fewer sample points than traditional methods. The central idea is to compute a Poisson-disk distribution of sample points for a small region of the screen (typically pixel-sized, or the size of a few pixels). Sample points are then assigned to pixels so that the density of samples points (rather than weights) for each pixel approximates a Gaussian (or other) reconstruction filter as closely as possible. The result is a supersampling kernel that implements importance sampling with Poisson-disk-distributed samples. The method incurs no additional run-time expense over standard weighted-average supersampling methods, supports successive-refinement, and can be implemented on any high-performance system that point samples accurately and has sufficient frame-buffer storage for two color buffers.
    [Show full text]
  • Lesson 22: Determining Control Stability Using Bode Plots
    11/30/2015 Lesson 22: Determining Control Stability Using Bode Plots 1 ET 438A AUTOMATIC CONTROL SYSTEMS TECHNOLOGY lesson22et438a.pptx Learning Objectives 2 After this presentation you will be able to: List the control stability criteria for open loop frequency response. Identify the gain and phase margins necessary for a stable control system. Use a Bode plot to determine if a control system is stable or unstable. Generate Bode plots of control systems the include dead-time delay and determine system stability. lesson22et438a.pptx 1 11/30/2015 Bode Plot Stability Criteria 3 Open loop gain of less than 1 (G<1 or G<0dB) at Stable Control open loop phase angle of -180 degrees System Oscillatory Open loop gain of exactly 1 (G=1 or G= 0dB) at Control System open loop phase angle of -180 degrees Marginally Stable Unstable Control Open loop gain of greater than 1 (G>1 or G>0dB) System at open loop phase angle of -180 degrees lesson22et438a.pptx Phase and Gain Margins 4 Inherent error and inaccuracies require ranges of phase shift and gain to insure stability. Gain Margin – Safe level below 1 required for stability Minimum level : G=0.5 or -6 dB at phase shift of 180 degrees Phase Margin – Safe level above -180 degrees required for stability Minimum level : f=40 degree or -180+ 40=-140 degrees at gain level of 0.5 or 0 dB. lesson22et438a.pptx 2 11/30/2015 Determining Phase and Gain Margins 5 Define two frequencies: wodB = frequency of 0 dB gain w180 = frequency of -180 degree phase shift Open Loop Gain 0 dB Gain Margin -m180 bodB Phase Margin -180+b0dB -180o Open Loop w w Phase odB 180 lesson22et438a.pptx Determining Phase and Gain Margins 6 Procedure: 1) Draw vertical lines through 0 dB on gain and -180 on phase plots.
    [Show full text]
  • EE C128 Chapter 10
    Lecture abstract EE C128 / ME C134 – Feedback Control Systems Topics covered in this presentation Lecture – Chapter 10 – Frequency Response Techniques I Advantages of FR techniques over RL I Define FR Alexandre Bayen I Define Bode & Nyquist plots I Relation between poles & zeros to Bode plots (slope, etc.) Department of Electrical Engineering & Computer Science st nd University of California Berkeley I Features of 1 -&2 -order system Bode plots I Define Nyquist criterion I Method of dealing with OL poles & zeros on imaginary axis I Simple method of dealing with OL stable & unstable systems I Determining gain & phase margins from Bode & Nyquist plots I Define static error constants September 10, 2013 I Determining static error constants from Bode & Nyquist plots I Determining TF from experimental FR data Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 1 / 64 Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 2 / 64 10 FR techniques 10.1 Intro Chapter outline 1 10 Frequency response techniques 1 10 Frequency response techniques 10.1 Introduction 10.1 Introduction 10.2 Asymptotic approximations: Bode plots 10.2 Asymptotic approximations: Bode plots 10.3 Introduction to Nyquist criterion 10.3 Introduction to Nyquist criterion 10.4 Sketching the Nyquist diagram 10.4 Sketching the Nyquist diagram 10.5 Stability via the Nyquist diagram 10.5 Stability via the Nyquist diagram 10.6 Gain margin and phase margin via the Nyquist diagram 10.6 Gain margin and phase margin via the Nyquist diagram 10.7 Stability, gain margin, and
    [Show full text]
  • I Introduction and Background
    Final Design Report Platform Motion Measurement System ELEC 492 A Senior Design Project, 2005 Between University of San Diego And Trex Enterprises Submitted to: Dr. Lord at USD Prepared by: YAZ Zlatko Filipovic Yoshitaka Yano August 12, 2005 University of San Diego Final Design Report USD August 12, 2005 Platform Motion Measurement System Table of Contents I. Acknowledgments…………...…………………………………………………………4 II. Executive Summary……..…………………………………………………………….5 III. Introduction and Background…...………………………...………………………….6 IV. Project Requirements…………….................................................................................8 V. Methodology of Design Plan……….…..……………..…….………….…..…….…..11 VI. Testing….……………………………………..…..…………………….……...........21 VII. Deliverables and Project Results………...…………...…………..…….…………...24 VIII. Budget………………………………………...……..……………………………..25 IX. Personnel…………………..………………………………………....….……….......28 X. Design Schedule………….…...……………………………….……...…………........29 XI. Reference and Bibliography…………..…………..…………………...…………….31 XII. Summary…………………………………………………………………………....32 Appendices Appendix 1. Simulink Simulations…..………………………….…….33 Appendix 2. VisSim Simulations..........……………………………….34 Appendix 3. Frequency Response and VisSim results……………......36 Appendix 4. Vibration Measurements…………..………………….….38 Appendix 5. PCB Layout and Schematic……………………………..39 Appendix 6. User’s Manual…………………………………………..41 Appendix 7. PSpice Simulations of Hfilter…………………………….46 1 Final Design Report USD August 12, 2005 Platform Motion Measurement System
    [Show full text]
  • Super-Sampling Anti-Aliasing Analyzed
    Super-sampling Anti-aliasing Analyzed Kristof Beets Dave Barron Beyond3D [email protected] Abstract - This paper examines two varieties of super-sample anti-aliasing: Rotated Grid Super- Sampling (RGSS) and Ordered Grid Super-Sampling (OGSS). RGSS employs a sub-sampling grid that is rotated around the standard horizontal and vertical offset axes used in OGSS by (typically) 20 to 30°. RGSS is seen to have one basic advantage over OGSS: More effective anti-aliasing near the horizontal and vertical axes, where the human eye can most easily detect screen aliasing (jaggies). This advantage also permits the use of fewer sub-samples to achieve approximately the same visual effect as OGSS. In addition, this paper examines the fill-rate, memory, and bandwidth usage of both anti-aliasing techniques. Super-sampling anti-aliasing is found to be a costly process that inevitably reduces graphics processing performance, typically by a substantial margin. However, anti-aliasing’s posi- tive impact on image quality is significant and is seen to be very important to an improved gaming experience and worth the performance cost. What is Aliasing? digital medium like a CD. This translates to graphics in that a sample represents a specific moment as well as a Computers have always strived to achieve a higher-level specific area. A pixel represents each area and a frame of quality in graphics, with the goal in mind of eventu- represents each moment. ally being able to create an accurate representation of reality. Of course, to achieve reality itself is impossible, At our current level of consumer technology, it simply as reality is infinitely detailed.
    [Show full text]
  • Meeting Design Specifications
    NDSU Meeting Design Specs using Bode Plots ECE 461 Meeting Design Specs using Bode Plots When designing a compensator to meet design specs including: Steady-state error for a step input Phase Margin 0dB gain frequency the procedure using Bode plots is about the same as it was for root-locus plots. Add a pole at s = 0 if needed (making the system type-1) Start cancelling zeros until the system is too fast For every zero you added, add a pole. Place these poles so that the phase adds up. When using root-locus techniques, the phase adds up to 180 degrees at some spot on the s-plane. When using Bode plot techniques, the phase adds up to give you your phase margin at some spot on the jw axis. Example 1: Problem: Given the following system 1000 G(s) = (s+1)(s+3)(s+6)(s+10) Design a compensator which result in No error for a step input 20% overshoot for a step input, and A settling time of 4 second Solution: First, convert to Bode Plot terminology No overshoot means make this a type-1 system. Add a pole at s = 0. 20% overshoot means ζ = 0.4559 Mm = 1.2322 GK = 1∠ − 132.120 Phase Margin = 47.88 degrees A settling time of 4 seconds means s = -1 + j X s = -1 + j2 (from the damping ratio) The 0dB gain frequency is 2.00 rad/sec In short, design K(s) so that the system is type-1 and 0 (GK)s=j2 = 1∠ − 132.12 JSG 1 rev June 22, 2016 NDSU Meeting Design Specs using Bode Plots ECE 461 Start with s+1 K(s) = s At 4 rad/sec 1000 0 GK = s(s+3)(s+6)(s+10) = 2.1508∠ − 153.43 s=j2 There is too much phase shift, so start cancelling zeros.
    [Show full text]
  • Designing Filters Using the Digital Filter Design Toolkit Rahman Jamal, Mike Cerna, John Hanks
    NATIONAL Application Note 097 INSTRUMENTS® The Software is the Instrument ® Designing Filters Using the Digital Filter Design Toolkit Rahman Jamal, Mike Cerna, John Hanks Introduction The importance of digital filters is well established. Digital filters, and more generally digital signal processing algorithms, are classified as discrete-time systems. They are commonly implemented on a general purpose computer or on a dedicated digital signal processing (DSP) chip. Due to their well-known advantages, digital filters are often replacing classical analog filters. In this application note, we introduce a new digital filter design and analysis tool implemented in LabVIEW with which developers can graphically design classical IIR and FIR filters, interactively review filter responses, and save filter coefficients. In addition, real-world filter testing can be performed within the digital filter design application using a plug-in data acquisition board. Digital Filter Design Process Digital filters are used in a wide variety of signal processing applications, such as spectrum analysis, digital image processing, and pattern recognition. Digital filters eliminate a number of problems associated with their classical analog counterparts and thus are preferably used in place of analog filters. Digital filters belong to the class of discrete-time LTI (linear time invariant) systems, which are characterized by the properties of causality, recursibility, and stability. They can be characterized in the time domain by their unit-impulse response, and in the transform domain by their transfer function. Obviously, the unit-impulse response sequence of a causal LTI system could be of either finite or infinite duration and this property determines their classification into either finite impulse response (FIR) or infinite impulse response (IIR) system.
    [Show full text]
  • Finite Impulse Response (FIR) Digital Filters (II) Ideal Impulse Response Design Examples Yogananda Isukapalli
    Finite Impulse Response (FIR) Digital Filters (II) Ideal Impulse Response Design Examples Yogananda Isukapalli 1 • FIR Filter Design Problem Given H(z) or H(ejw), find filter coefficients {b0, b1, b2, ….. bN-1} which are equal to {h0, h1, h2, ….hN-1} in the case of FIR filters. 1 z-1 z-1 z-1 z-1 x[n] h0 h1 h2 h3 hN-2 hN-1 1 1 1 1 1 y[n] Consider a general (infinite impulse response) definition: ¥ H (z) = å h[n] z-n n=-¥ 2 From complex variable theory, the inverse transform is: 1 n -1 h[n] = ò H (z)z dz 2pj C Where C is a counterclockwise closed contour in the region of convergence of H(z) and encircling the origin of the z-plane • Evaluating H(z) on the unit circle ( z = ejw ) : ¥ H (e jw ) = åh[n]e- jnw n=-¥ 1 p h[n] = ò H (e jw )e jnwdw where dz = jejw dw 2p -p 3 • Design of an ideal low pass FIR digital filter H(ejw) K -2p -p -wc 0 wc p 2p w Find ideal low pass impulse response {h[n]} 1 p h [n] = H (e jw )e jnwdw LP ò 2p -p 1 wc = Ke jnwdw 2p ò -wc Hence K h [n] = sin(nw ) n = 0, ±1, ±2, …. ±¥ LP np c 4 Let K = 1, wc = p/4, n = 0, ±1, …, ±10 The impulse response coefficients are n = 0, h[n] = 0.25 n = ±4, h[n] = 0 = ±1, = 0.225 = ±5, = -0.043 = ±2, = 0.159 = ±6, = -0.053 = ±3, = 0.075 = ±7, = -0.032 n = ±8, h[n] = 0 = ±9, = 0.025 = ±10, = 0.032 5 Non Causal FIR Impulse Response We can make it causal if we shift hLP[n] by 10 units to the right: K h [n] = sin((n -10)w ) LP (n -10)p c n = 0, 1, 2, ….
    [Show full text]
  • Applications Academic Program Distributing Vissim Models
    VISSIM VisSim is a visual block diagram language for simulation of dynamical systems and Model Based Design of embedded systems. It is developed by Visual Solutions of Westford, Massachusetts. Applications VisSim is widely used in control system design and digital signal processing for multidomain simulation and design. It includes blocks for arithmetic, Boolean, and transcendental functions, as well as digital filters, transfer functions, numerical integration and interactive plotting. The most commonly modeled systems are aeronautical, biological/medical, digital power, electric motor, electrical, hydraulic, mechanical, process, thermal/HVAC and econometric. Academic program The VisSim Free Academic Program allows accredited educational institutions to site license VisSim v3.0 for no cost. The latest versions of VisSim and addons are also available to students and academic institutions at greatly reduced pricing. Distributing VisSim models VisSim viewer screenshot with sample model. The free Vis Sim Viewer is a convenient way to share VisSim models with colleagues and clients not licensed to use VisSim. The VisSim Viewer will execute any VisSim model, and allows changes to block and simulation parameters to illustrate different design scenarios. Sliders and buttons may be activated if included in the model. Code generation The VisSim/C-Code add-on generates efficient, readable ANSI C code for algorithm acceleration and real-time implementation of embedded systems. The code is more efficient and readable than most other code generators. VisSim's author served on the X3J11 ANSI C committee and wrote several C compilers, in addition to co-authoring a book on C. [2] This deep understanding of ANSI C, and the nature of the resulting machine code when compiled, is the key to the code generator's efficiency.
    [Show full text]
  • Efficient Multidimensional Sampling
    EUROGRAPHICS 2002 / G. Drettakis and H.-P. Seidel Volume 21 (2002 ), Number 3 (Guest Editors) Efficient Multidimensional Sampling Thomas Kollig and Alexander Keller Department of Computer Science, Kaiserslautern University, Germany Abstract Image synthesis often requires the Monte Carlo estimation of integrals. Based on a generalized con- cept of stratification we present an efficient sampling scheme that consistently outperforms previous techniques. This is achieved by assembling sampling patterns that are stratified in the sense of jittered sampling and N-rooks sampling at the same time. The faster convergence and improved anti-aliasing are demonstrated by numerical experiments. Categories and Subject Descriptors (according to ACM CCS): G.3 [Probability and Statistics]: Prob- abilistic Algorithms (including Monte Carlo); I.3.2 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism. 1. Introduction general concept of stratification than just joining jit- tered and Latin hypercube sampling. Since our sam- Many rendering tasks are given in integral form and ples are highly correlated and satisfy a minimum dis- usually the integrands are discontinuous and of high tance property, noise artifacts are attenuated much 22 dimension, too. Since the Monte Carlo method is in- more efficiently and anti-aliasing is improved. dependent of dimension and applicable to all square- integrable functions, it has proven to be a practical tool for numerical integration. It relies on the point 2. Monte Carlo Integration sampling paradigm and such on sample placement. In- The Monte Carlo method of integration estimates the creasing the uniformity of the samples is crucial for integral of a square-integrable function f over the s- the efficiency of the stochastic method and the level dimensional unit cube by of noise contained in the rendered images.
    [Show full text]
  • Frequency Response and Bode Plots
    1 Frequency Response and Bode Plots 1.1 Preliminaries The steady-state sinusoidal frequency-response of a circuit is described by the phasor transfer function Hj( ) . A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. Of course we can easily program the transfer function into a computer to make such plots, and for very complicated transfer functions this may be our only recourse. But in many cases the key features of the plot can be quickly sketched by hand using some simple rules that identify the impact of the poles and zeroes in shaping the frequency response. The advantage of this approach is the insight it provides on how the circuit elements influence the frequency response. This is especially important in the design of frequency-selective circuits. We will first consider how to generate Bode plots for simple poles, and then discuss how to handle the general second-order response. Before doing this, however, it may be helpful to review some properties of transfer functions, the decibel scale, and properties of the log function. Poles, Zeroes, and Stability The s-domain transfer function is always a rational polynomial function of the form Ns() smm as12 a s m asa Hs() K K mm12 10 (1.1) nn12 n Ds() s bsnn12 b s bsb 10 As we have seen already, the polynomials in the numerator and denominator are factored to find the poles and zeroes; these are the values of s that make the numerator or denominator zero. If we write the zeroes as zz123,, zetc., and similarly write the poles as pp123,, p , then Hs( ) can be written in factored form as ()()()s zsz sz Hs() K 12 m (1.2) ()()()s psp12 sp n 1 © Bob York 2009 2 Frequency Response and Bode Plots The pole and zero locations can be real or complex.
    [Show full text]