This document provides an overview of digital filters, including common filter types, difference equations, and diagram representations. It discusses low pass, high pass, band pass, and band stop filters. Filters are defined by their frequency response and can be implemented using difference equations in either recursive or non-recursive form. Difference equations can be represented using diagrams with delay, coefficient multiplication, and summer elements. The impulse response and step response are also introduced as ways to characterize a filter or system.
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Nhận dạng mặt người bằng thuật toán PCA trên Matlabhieu anh
Mục tiêu của đề tài “ Nhận dạng mặt người bằng thuật toán PCA trên Matlab ” là thực hiện chương trình tìm kiếm một bức ảnh có khuôn mặt một người trong tập ảnh cơ sở giống với khuôn mặt của người trong bức ảnh cần kiểm tra bằng ngôn ngữ Matlab.
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Nhận dạng mặt người bằng thuật toán PCA trên Matlabhieu anh
Mục tiêu của đề tài “ Nhận dạng mặt người bằng thuật toán PCA trên Matlab ” là thực hiện chương trình tìm kiếm một bức ảnh có khuôn mặt một người trong tập ảnh cơ sở giống với khuôn mặt của người trong bức ảnh cần kiểm tra bằng ngôn ngữ Matlab.
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Phương pháp sai phân hữu hạn và phần tử hữu hạn trong truyền nhiệtTrinh Van Quang
Bài giảng cho các lớp cao học cơ khí. Nội dung trình bày các kiến thức cơ bản của phương pháp sai phân hữu hạn và phần tử hữu hạn trong truyền nhiệt. Tài liệu cũng có thể được tham khảo để áp dụng hai phương pháp này trong nghiên cứu tính nhiệt công trình, vật thể.
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This document discusses digital IIR filter design. It describes different types of analog filters like Butterworth, Chebyshev, Elliptic and Bessel filters. It then explains how to design digital IIR filters using analog filter design principles with impulse invariant and bilinear transformations from the s-domain to the z-domain. Step-by-step procedures are provided for designing low-pass Butterworth and Chebyshev filters using these two transformation methods. The concepts of aliasing and pre-warping frequencies are also covered in the bilinear transformation section.
This document discusses the design of IIR and FIR filters. IIR (Infinite Impulse Response) filters are analog filters that use feedback and have non-linear phase responses. Common IIR design methods are impulse invariant, bilinear transformation, and approximation of derivatives. FIR (Finite Impulse Response) filters are digital filters with no feedback and linear phase responses. FIR filters are designed using windowing methods like rectangular, Hamming, and Kaiser windows which concentrate the filter response around the desired frequencies. IIR filters require less computation but FIR filters are required where linear phase response is needed such as data transmission and speech processing.
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Phương pháp sai phân hữu hạn và phần tử hữu hạn trong truyền nhiệtTrinh Van Quang
Bài giảng cho các lớp cao học cơ khí. Nội dung trình bày các kiến thức cơ bản của phương pháp sai phân hữu hạn và phần tử hữu hạn trong truyền nhiệt. Tài liệu cũng có thể được tham khảo để áp dụng hai phương pháp này trong nghiên cứu tính nhiệt công trình, vật thể.
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This document discusses digital IIR filter design. It describes different types of analog filters like Butterworth, Chebyshev, Elliptic and Bessel filters. It then explains how to design digital IIR filters using analog filter design principles with impulse invariant and bilinear transformations from the s-domain to the z-domain. Step-by-step procedures are provided for designing low-pass Butterworth and Chebyshev filters using these two transformation methods. The concepts of aliasing and pre-warping frequencies are also covered in the bilinear transformation section.
This document discusses the design of IIR and FIR filters. IIR (Infinite Impulse Response) filters are analog filters that use feedback and have non-linear phase responses. Common IIR design methods are impulse invariant, bilinear transformation, and approximation of derivatives. FIR (Finite Impulse Response) filters are digital filters with no feedback and linear phase responses. FIR filters are designed using windowing methods like rectangular, Hamming, and Kaiser windows which concentrate the filter response around the desired frequencies. IIR filters require less computation but FIR filters are required where linear phase response is needed such as data transmission and speech processing.
This document discusses multirate digital signal processing and basic sampling rate alteration devices. It describes up-samplers and down-samplers in the time and frequency domains. Up-samplers increase the sampling rate by inserting zeros, which in the frequency domain causes images of the input spectrum. Down-samplers decrease the sampling rate by selecting samples, which can cause aliasing due to spectrum overlap if the Nyquist criterion is not met. The time-varying and frequency translation properties of up-samplers and down-samplers are illustrated through examples.
1. Combinational Logic Circutis with examples (1).pdfRohitkumarYadav80
This document provides an overview of combinational circuits including adders, subtractors, and code converters. It discusses the design process for combinational circuits and considerations like minimizing gates and propagation time. Specific circuit components are then explained, including half adders, full adders, half subtractors, and full subtractors. Their truth tables and logic diagrams are presented. Finally, code converters are briefly mentioned as an application of combinational circuits.
1. Combinational Logic Circutis with examples (1).pdfRohitkumarYadav80
The document discusses combinational circuits including adders, subtractors, and code converters. It provides details on the design process for combinational circuits including determining inputs and outputs, deriving truth tables, obtaining Boolean functions, and drawing logic diagrams. Specific circuit components are then covered in more depth, including half adders, full adders, half subtractors, and full subtractors. Their definitions, procedures, truth tables, and logic diagrams are presented. Finally, code converters are briefly mentioned as an application of combinational circuits.
Description about -
1.filter and types of filter
2.comb filter defination & its magnitude and phase response
3.Digital comb filter defination & its magnitude and phase response
4.Digital comb filter using a digital differentiator & its magnitude and phase response
5.Comb Filters with Multiple Delay Elements defination & its magnitude and phase response
6.Digital Integrator(non-delaying) defination & its magnitude and phase response
7.Delaying Integrator defination & its magnitude and phase response
8.Fourier Transform & Dirac Delta Function (Unit Impulse Response) and its properties
The document discusses techniques for designing discrete-time infinite impulse response (IIR) filters from continuous-time filter specifications. It covers the impulse invariance method, matched z-transform method, and bilinear transformation method. The impulse invariance method samples the continuous-time impulse response to obtain the discrete-time impulse response. The bilinear transformation maps the entire s-plane to the unit circle in the z-plane to avoid aliasing. Examples are provided to illustrate the design process using each method.
This document discusses adaptive filters and band reject filters. It begins by defining an adaptive filter as a filter that can automatically design itself and detect system variations over time. It describes the key aspects of an adaptive filter including the signals being processed, its structure, adjustable parameters, and adaptive algorithm. The document then discusses practical adaptive filtering problems when the desired response is unknown and time-varying. It introduces common adaptive filtering algorithms like the gradient-based LMS algorithm and examines the mean-squared error cost function. Finally, it provides an overview of band reject filters and inverse and Wiener filtering techniques.
Digital filters can remove unwanted noise from signals or extract useful frequency components. They operate by sampling an analog signal, processing the digital values, and converting back to analog. Finite impulse response (FIR) filters use weighted sums of past inputs for outputs and are inherently stable without feedback. Infinite impulse response (IIR) filters use feedback, with outputs and next states determined by inputs and past outputs. Common filters include moving average filters and filters that introduce gain, delay, or differences between signal values. Design involves selecting coefficients for desired frequency responses. Stability depends on pole locations within the unit circle. Digital filters find applications in communications, audio, imaging, and other areas.
1. The document discusses filters, including analog filters that process continuous-time signals and digital filters that process discrete-time signals. It also discusses different types of filters like lowpass, highpass, bandstop, and bandpass filters.
2. Active filters are described as overcoming some of the drawbacks of passive filters by using op-amps instead of inductors. While they have some disadvantages like limited bandwidth, active filters are commonly used for voice and data communications due to economic and performance advantages over passive filters.
3. Bode plots are introduced as important tools for analyzing the frequency response of filters by plotting magnitude or phase versus frequency on logarithmic scales. Corner frequency is defined as the frequency where the
The document discusses digital filters and their design. It begins with an introduction to filters and their uses in signal processing applications. It then covers linear time-invariant filters and their transfer functions. It discusses the differences between non-recursive (FIR) and recursive (IIR) filters. The document presents various filter structures for implementation, including direct form I and direct form II structures. It also discusses designing FIR and IIR filters as well as issues in their implementation.
This document describes various types of combinational logic circuits. It discusses half adders and full adders which can add single bits and pairs of bits along with a carry bit. Parallel adders are formed from cascaded full adders to add multiple bit numbers. Half subtractors, full subtractors, and parallel subtractors can perform single and multi-bit binary subtraction. Multiplexers and demultiplexers are described which can select single inputs from multiple options or distribute a single input to multiple outputs. Encoders and decoders are covered which convert between binary and coded representations.
This document discusses filter implementation and the finite word length problem. It covers topics like coefficient quantization, arithmetic operations, quantization noise, and limit cycles. Coefficient quantization can shift pole and zero locations from their ideal positions. Statistical analysis models quantization noise as independent white noise sources, while deterministic analysis may reveal limit cycles due to non-linear quantization effects. Different realizations like direct form and lattice structures are also compared in terms of their sensitivity to finite word length issues.
1. The document discusses multirate signal processing and the effects of finite word length. It covers topics like downsampling, upsampling, decimation filters, interpolation filters, and aliasing.
2. Finite word length effects cause errors from input quantization, coefficient quantization, and truncated product terms. This results in noise and a reduction in signal-to-noise ratio.
3. With finite precision, systems can exhibit limit cycles where the output assumes a repeating set of values within a "deadband". This emulates instability in continuous systems.
Linear programming is a technique for choosing the best alternative from a set of feasible alternatives by expressing the objective function and constraints as linear mathematical functions. It requires a clear objective, quantitative terms, identifiable constraints, and feasible alternative choices. The objective functions and constraints define a feasible region, and the simplex method is an iterative algorithm that generates corner point solutions to find the optimal solution at an extreme point of the feasible region. It works by maintaining a basic feasible solution and performing elementary row operations at each iteration to improve the objective function value until reaching the optimal solution.
This document summarizes techniques for designing finite impulse response (FIR) filters and digital controllers. It discusses truncating the impulse response of an ideal filter to design FIR filters. It also presents a method for digital controller design that matches the step response of an analog controller to approximate it digitally. Examples are provided of designing an FIR lowpass filter and approximating an analog feedback controller with a digital controller using step response matching.
This document summarizes techniques for designing finite impulse response (FIR) filters and digital controllers. It discusses truncating the impulse response of an ideal filter to design FIR filters. It also presents a method for digital controller design that matches the step response of an analog controller to approximate it digitally. Examples are provided of designing an FIR lowpass filter and approximating an analog feedback controller with a digital controller using step response matching.
The document presents a new hybrid method for performing binary floating point multiplication that aims to improve speed. It combines the Dadda multiplier and modified radix-8 Booth multiplier algorithms. The hybrid method multiplies the mantissas using this approach, replacing existing multiplier designs like carry save multipliers. It achieves a maximum frequency of 555MHz, faster than existing floating point multipliers. The design is implemented in Verilog HDL and tested using Quartus II software.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
2. Overview
• Filters
• Common Filter Types
• Analog vs. Digital Filters
• Difference Equation
• Difference Equation Diagram Elements
• Nonrecursive Difference Equation Diagrams
• Recursive Difference Equation Diagrams
• Impulse Response
• Step Response
3. Filters
• Filters change a signal’s characteristics by selectively removing
some of its frequency elements.
• A low pass filter, for example, removes the high frequency
components of a signal, but passes low frequency
components.
• It does this because the filter’s gain, the amplification factor it
applies to an input, varies with frequency.
3
4. Filters
• For a low pass filter, the gains are highest at low frequency
and much lower at high frequency.
• A high pass filter has exactly the opposite shape: its gains are
highest at high frequency and lowest at low frequency.
• The pass band of a filter determines the range of frequencies
that are passed.
• The stop band of the filter determines the range of
frequencies that are strongly attenuated.
4
5. Filters
Cutoff Frequency
• A filter is considered to pass signals at frequencies where the filter’s
gain exceeds 0.707 of its maximum gain.
• The frequency or frequencies where the gain equals 0.707 of the
maximum gain are called the cut-off frequencies of the filter.
• Since 20log0.707 = -3 dB, they are also called –3 dB frequencies.
5
6. Filters
Bandwidth
• The bandwidth of a low pass filter is the range of frequencies from
0 to the –3 dB frequency.
• For a high pass filter, the bandwidth is the range of frequencies
from the –3 dB frequency to half the sampling frequency.
• For band pass filters, the bandwidth is the distance in Hz between
the cut-off frequencies.
6
11. Filters
Roll-Off
• Another important feature of a filter is its roll-off.
• This characteristic determines how quickly the gain drops
outside the pass band.
• The higher the order of a digital filter, defined by the number
of coefficients needed to specify it, the steeper the slope, and
the higher the quality of the filter is said to be.
11
12. Filters
Gain
• A filter’s gain at a certain frequency determines the
amplification factor that the filter applies to an input at this
frequency.
• A gain may have any value.
• In the pass band region, filter’s gain is high.
• In the stop band region, filter’s gain is low.
12
13. Filters
Gain
• The cutoff frequency of the filter occurs when the gain is
1
2
~0.707 𝑜𝑟 70.7%
• A gain in dB is calculated as 𝐺𝑎𝑖𝑛 𝑑𝐵 = 20𝑙𝑜𝑔 𝐺𝑎𝑖𝑛
13
14. Filters
Bandwidth
Example-1: Find the bandwidth of the band pass filter.
The edges of the pass
Band occur where the
Gain equals 0.707.
The bandwidth of the
Filter is
BW = FH - FL
BW = 4000 - 2000
BW = 2000 Hz
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15. Square Wave
• A square wave can be constructed from multiple sine waves at different
frequencies.
• The sine waves added in addition to the fundamental frequency are
called harmonics.
• A square wave has harmonics at odd multiples of the fundamental
frequency.
15
16. Square Wave
• As higher harmonics are added, the result gets closer to an ideal square
wave, which contains infinite harmonics.
• sin(angle) + sin(3*angle)/3 + sin(5*angle)/5 + sin(7*angle)/7 + ...
16
17. Effects of Low Pass Filters
17
• Each filter type has a unique effect on an input signal.
• Low pass filters tend to smooth signals by averaging out sudden changes.
18. Effects of High Pass Filters
18
• Each filter type has a unique effect on an input signal.
• High pass filters tend emphasize sharp transition.
20. Analog vs. Digital Filters
• Filters may be implemented in either analog or digital form.
• Analog filters are defined in hardware, while digital filters are
defined in software.
• In general, digital filters are much less susceptible to noise
and component variation than are analog filters.
• Furthermore, digital filter re-design is simply a matter of
editing a list of coefficients, while analog re-design requires
building an entirely new circuit.
20
21. Difference Equation
• A difference equation is one way to specify a filter or system.
• It may be presented in equation or diagram form.
• The general form of a difference equation is
a0y[n] + a1y[n-1] + a2y[n-2] + … + aNy[n-N]
= b0x[n] + b1x[n-1] + b2x[n-2] + … + bMx[n-M]
• The ak and bk weightings are called filter coefficients.
• Generally difference equation has N+1 ak coefficients and M+1 bk
coefficients.
• N is the past output required, also referred as a filter order.
• M is the number of past input required.
• The equation defines how each new output y[n] is obtained.
• In general, a0 is assumed to be one.
21
22. Difference Equation
• The general form of a difference equation is
a0y[n] + a1y[n-1] + a2y[n-2] + … + aNy[n-N]
= b0x[n] + b1x[n-1] + b2x[n-2] + … + bMx[n-M]
The compact form of the recursive difference equation is
• In the above equation, both inputs and outputs are needed to
compute a new output, so the difference equation is said to
be recursive.
22
23. Difference Equation
• When only inputs are needed to compute a new output, the
difference equation is said to be non-recursive, that is,
y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + … + bMx[n-M]
The compact form of the non-recursive difference equation is
23
24. Difference Equation
Example-2: A filter has a difference equation y[n] = 0.5y[n-1] + x[n]
a. Is this a recursive or non-recursive difference equation?
b. Identify all ak and bk coefficients.
c. If the input x[n] is u[n], find 5 samples of the output y[n].
Solution
a. The equation is recursive as it depends on a past output y[n-1]
b. Since y[n] – 0.5y[n-1] = x[n], therefore a0 = 1, a1 = -0.5, b0 = 1
c. y[n] = 0.5y[n-1] + x[n]
When n = 0, y[0] = 0.5y[0-1] + x[0] = 0.5(0) + 1 = 1
When n = 1, y[1] = 0.5y[1-1] + x[1] = 0.5(1) + 1 = 1.5
When n = 2, y[2] = 0.5y[2-1] + x[2] = 0.5(1.5) + 1 = 1.75
When n = 3, y[3] = 0.5y[3-1] + x[3] = 0.5(1.75) + 1 = 1.875
When n = 4, y[4] = 0.5y[4-1] + x[4] = 0.5(1.875) + 1 = 1.9375
24
Y[n] = {1, 1.5, 1.75, 1.875, 1.9375}
29. Nonrecursive Difference Equation Diagram
• The general form of the nonrecursive difference equation is
y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + … + bMx[n-M]
• It can be schematically presented as
29
30. Finite Word Length Effects
• When difference equations are implemented by a computer,
quantization means that filter coefficients cannot be represented
perfectly.
• This means that filter behavior will not match the design exactly.
• Furthermore, computer arithmetic is always subject to rounding
and truncation errors, again due to the limited number of bits
available.
• All of these effects are called finite word length effects.
30
31. Nonrecursive Difference Equation Diagram
• In order to reduce quantization error of filter’s coefficient one
strategy is to break higher order filter into second order chunks
31
Figure: Cascaded second order nonrecursive filter sections
32. Nonrecursive Difference Equation Diagram
Example-3: Draw a diagram for the difference equation
y[n] = 0.5x[n] + 0.4x[n-1] - 0.2x[n-2]
Solution
32
33. Nonrecursive Difference Equation Diagram
Example-4: Write the difference equation of the following diagram.
Solution
y[n] = x[n] - 0.3x[n-2] + 0.7x[n-3] 33
34. Recursive Difference Equation Diagram
• The general form of a recursive difference equation is
a0y[n] + a1y[n-1] + a2y[n-2] + … + aNy[n-N]
= b0x[n] + b1x[n-1] + b2x[n-2] + … + bMx[n-M]
34
35. Recursive Difference Equation Diagram
Example-5: Draw a direct form I diagram for the following
recursive difference equation
y[n] + 0.5y[n-2] = 0.8x[n] + 0.1x[n-1] - 0.3x[n-2]
Solution
35
36. Recursive Difference Equation Diagram
Example-6: Write the difference equation of the following diagram.
Solution
y[n] + 0.5y[n-2] = 0.8x[n] + 0.1x[n-1] - 0.3x[n-2]
36
37. Recursive Difference Equation Diagram
• In direct form I realization, calculation for output y[n] require
M + 1 input states
N output states
M + N + 1 coefficient multiplications
M + N additions
• When more than 2 or 3 delays are needed, this realization is
very sensitive to the finite word length effects.
37
38. Recursive Difference Equation Diagram
• Recursive filters have an alternative representation, called direct form 2,
that has certain advantages for implementation.
• This representation is defined by the pair of equations
w[n] = x[n] – a1w[n-1] – a2w[n-2] – … – aNw[n-N]
y[n] = b0w[n] + b1w[n-1] + b2w[n-2] + … + bMw[n-M]
38
39. Recursive Difference Equation Diagram
• Transpose of the direct form 2 realization is an another
popular implementation model.
39
40. Recursive Difference Equation Diagram
Example-7: Write the difference equation of the following diagram.
• The output of the bottom summer is 0.1x[n] – 0.3y[n]
• The output of the middle summer is 0.1x[n-1] – 0.3y[n-1] + 0.2x[n] – 0.2y[n]
• The final out top summer is y[n] = 0.1x[n-2] – 0.3y[n-2] + 0.2x[n-1] – 0.2y[n-1] + 0.8x[n]
40
41. Impulse Response
• The impulse response h[n] is an important way of characterizing a
filter or system.
• By definition, it is the system’s response to an impulse function
input δ[n].
41
42. Impulse Response
• For a musical instrument like a piano, an impulse response
corresponds to the note obtained by striking a single key.
42
43. Impulse Response
• The impulse response may be calculated from a system’s
difference equation.
• The impulse function δ[n] substitutes for the input x[n]
• The impulse response h[n] substitutes for the output y[n]
43
44. Impulse Response
• For a non-recursive system, the impulse response is
h[n] = b0δ[n] + b1δ[n-1] + b2δ[n-2] + … + bMδ[n-M]
• For a recursive system, the impulse response is
h[n] = – a1h[n-1] – a2h[n-2] – … – aNh[n-N]
+ b0δ[n] + b1δ[n-1] + b2δ[n-2] + … + bMδ[n-M]
44
45. Impulse Response
• Because non-recursive systems do not rely on past outputs,
they have finite impulse responses, which return to zero after
a finite number of samples have elapsed.
• Recursive systems, on the other hand, have infinite impulse
responses, because each new output depends on past
outputs as well as inputs.
• For causal systems, both non-recursive and recursive, the
impulse response h[n] is zero for n < 0.
45
46. Impulse Response
Example-8: Find the first six samples of the impulse response
h[n] for the difference equation
Y[n] – 0.4y[n-1] = x[n] – x[n-1]
Solution
The impulse response of the difference equation is
h[n] – 0.4h[n-1] = δ[n] – δ[n-1]
So the first six samples of the impulse response are
h[0] = 1 h[3] = -0.096
h[1] = -0.6 h[4] = -0.0384
h[2] = -0.24 h[5] = -0.01536
46
47. Impulse Response
So the first six samples of the impulse response are
h[0] = 1 h[3] = -0.096
h[1] = -0.6 h[4] = -0.0384
h[2] = -0.24 h[5] = -0.01536
h[n] = [1, 0.6, -0.24, -0.096, -0.0384, -0.01536]
47
48. Impulse Response
Example-9: write the difference equation whose impulse
response is shown in the figure.
Solution
The impulse response can be written as a sum of impulse
functions.
h[n] = δ[n] + 0.8δ[n-1] + 0.2δ[n-2]
The difference equation is y[n] = x[n] + 0.8x[n-1] + 0.2x[n-2]
48
49. Impulse Response
Example-10: The input signal x[n] and impulse response h[n] are
shown in the figures. Find the output y[n] by breaking the input
signal into impulse functions and finding the response to each.
49
51. Step Response
• The step response s[n] for a filter is its response to a step
input u[n].
• It records the response of the system to a change in level, and
may be calculated in two ways.
• The first way mimics that for the impulse response, but with
s[n] replacing y[n] and u[n] replacing x[n].
51
53. Step Response
• The step function u[n] is equivalent to a sum of impulse
functions:
u[n] = δ[n] + δ[n-1] + δ[n-2] + …
• The step response s[n] is the same sum of impulse responses:
s[n] = h[n] + h[n-1] + h[n-2] + …
53
54. Step Response
• The second way to compute the step response s[n] relies on
the impulse response.
• each step response sample is a cumulative sum of the impulse
response samples.
• For example
s[4] = h[4] + h[3] + h[2] + h[1] + h[0]
54
55. Step Response
Example-11: Find the first eight samples of the step response
s[n] for the difference equation
Y[n] – 0.2y[n-1] = 0.5x[n] + 0.3x[n-1]
Solution
The step response s[n] of the difference equation is
s[n] – 0.4s[n-1] = u[n] – u[n-1]
So the first eight samples of the step response are
s[0] = 0.5 s[3] = 0.96 s[6] = 0.996
s[1] = 0.8 s[4] = 0.98 s[7] = 0.9984
s[2] = 0.9 s[5] = 0.992
55
56. Step Response
So the first eight samples of the step response are
s[0] = 0.5 s[3] = 0.96 s[6] = 0.996
s[1] = 0.8 s[4] = 0.98 s[7] = 0.9984
s[2] = 0.9 s[5] = 0.992
S[n] = [0.5, 0.8, 0.9, 0.96, 0.98, 0.992, 0.996, 0.9984]
56