Difference equation.pptx
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This document discusses linear constant coefficient difference equations, which can represent an important class of linear time-invariant (LTI) systems. It provides examples of representing systems as difference equations, such as an accumulator system. It also describes recursively computing the output of a difference equation and solving difference equations, noting that auxiliary conditions are needed to uniquely specify the output. If the additional condition is that the system starts at rest, then the system will be linear, time-invariant, and causal.
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![© Copyright 2014 Dawar Awan
CECOS College of Engineering and IT dawar@cecos.edu.pk
What is Digital Signal Processor ?
Linear Constant Coefficient Difference Equation
2
│ Not all, but, an important class of LTI systems can be represented in the form of
the following difference equation
│ Any system represented as the difference equation is not necessarily an LTI
system (it depends on auxiliary conditions)
│ The above mentioned equation is known as Nth order linear constant coefficient
difference equation
𝑘=0
𝑁
𝑎𝑘𝑦 𝑛 − 𝑘 =
𝑚=0
𝑀
𝑏𝑚𝑥[𝑛 − 𝑚]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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3 Technion – Israel Institute of Technology
4 Imperial College London
Types of workloads k 04tow
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Difference equation.pptx
- 1. Linear Constant Coefficient Difference Equation 1
- 2. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Linear Constant Coefficient Difference Equation 2 │ Not all, but, an important class of LTI systems can be represented in the form of the following difference equation │ Any system represented as the difference equation is not necessarily an LTI system (it depends on auxiliary conditions) │ The above mentioned equation is known as Nth order linear constant coefficient difference equation 𝑘=0 𝑁 𝑎𝑘𝑦 𝑛 − 𝑘 = 𝑚=0 𝑀 𝑏𝑚𝑥[𝑛 − 𝑚]
- 3. │ Consider an accumulator system © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Representing a System as a Difference Equation 3 𝑦 𝑛 = 𝑘=−∞ 𝑛 𝑥[𝑘] 𝑦 𝑛 − 1 = 𝑘=−∞ 𝑛−1 𝑥[𝑘] 𝑦 𝑛 = 𝑥 𝑛 + 𝑘=−∞ 𝑛−1 𝑥[𝑘] 𝑦 𝑛 = 𝑥 𝑛 + 𝑦[𝑛 − 1] 𝑦 𝑛 − 𝑦[𝑛 − 1] = 𝑥 𝑛 Structure for implementation of accumulator
- 4. │ Recursive computation of a difference equation © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 4
- 5. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 5 𝑎 𝑎 2 𝑐 + 𝑎𝑘 + 0
- 6. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 6
- 7. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 7
- 8. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 8 │ Proving the time invariance of the system
- 9. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 9 │ The output for a given input is not uniquely specified, auxiliary information or conditions are required. │ If the auxiliary information is in the form of 𝑁 sequential values of the output, later values can be obtained by rearranging the difference equation as a recursive relation running forward in 𝑛, and prior values can be obtained by rearranging the difference equation as a recursive relation running backward in 𝑛. │ Linearity, time invariance, and causality of the system will depend on the auxiliary conditions. If the additional condition is that the system is initially at rest, then the system will be linear, time invariant, and causal.
- 10. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 10 │ Now considering initial rest condition i.e │ 𝑥 𝑛 = 0 𝑓𝑜𝑟 𝑛 < 0 𝑎𝑛𝑑 𝑠𝑜 𝑦 𝑛 = 0 𝑓𝑜𝑟 𝑛 < 0
- 11. © Copyright 2014 Dawar Awan CECOS College of Engineering and IT dawar@cecos.edu.pk What is Digital Signal Processor ? Solving a Difference Equation 11