EC-7.7 Discrete-Time Systems and the Difference Equation
You can write and solve a difference equation, find the transfer function of a discrete-time system, decide its stability, and draw it as a block diagram.
Before:EC-4. Transforms, Complex Variables and Numerical MethodsUnlocks:S4-1. Analog CircuitsS4-3. Control SystemsS4-5. Probability Theory & Stochastic ProcessesS5-3. Digital Signal ProcessingPE2-3. Biomedical Instrumentation and Signal ProcessingEC-9. Microcontrollers and Embedded Systems
Everything a processor does to a signal is a difference equation, so this is the topic that connects the whole module to code. It is also the exit point of the module: digital signal processing starts here and spends its time on choosing coefficients, which this topic deliberately does not do. What it does instead is make sure a reader arriving at that subject already knows what the coefficients are coefficients of.
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The difference equation, and what it stands in for
A discrete-time system is described by relating the present output to past outputs and to present and past inputs. It plays the role the differential equation plays in continuous time, and it is far easier to compute with.
NPTEL: Discrete Time Signal Processing · CourseSolving one by stepping forward, and the two parts of the answer
Given the coefficients, the initial values and the input, the output follows one position at a time. The result separates into a part set by the initial values and a part set by the input, and only the second survives in the long run.
NPTEL: Principles of Signals and Systems · CourseThe transfer function in the z domain, and poles inside the unit circle
Transforming the difference equation gives a ratio of polynomials again, and a causal discrete-time system is stable when every pole lies strictly inside the unit circle. It is the same argument as before with a different boundary.
NPTEL: Discrete Time Signal Processing · CourseFinite and infinite impulse response, and what the difference costs
A system with no feedback path has an impulse response of finite length and cannot be unstable; one with feedback can be far more efficient and can also be unstable. The choice between them is a real engineering trade.
NPTEL: Signals and Systems (IISER Bhopal) · CourseBlock diagrams made of delays, gains and adders
Any difference equation can be drawn with three elements, and the drawing is close to how it would be written in code or laid out in hardware. Reading one back into an equation is the check that you understand it.
The frequency response of a discrete-time system, and why it repeats
Evaluating the transfer function on the unit circle gives the frequency response, and it necessarily repeats because the sampled description cannot distinguish frequencies that differ by the sampling rate. The previous topic is why.
NPTEL: Signals and Systems (IIT Kanpur) · Course
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