EC-7.7 Discrete-Time Systems and the Difference Equation

What this is and why it exists

Everything a processor does to a signal is a difference equation. That is not an analogy. A filter running on a chip is a few lines of arithmetic relating present output to past output and to present and past input. That arithmetic is what this topic is about, so this is where the module connects to code.

It is also the module's exit. Digital signal processing starts exactly here, and then spends its time on a question this topic does not touch. How should the coefficients be chosen so the response meets a specification? That question needs a course.

What this topic does instead is make sure a reader arriving at that course already knows what the coefficients are coefficients of. Such a reader can also tell whether a given set of them will blow up.

The vocabulary

• Difference equation — a relation expressing the present output in terms of past outputs and present and past inputs.
• Order — how far back the equation reaches.
• Recursive — using past output values, which means feedback.
• Non-recursive — using only input values, which means no feedback.
• Finite impulse response — an impulse response of finite length. Non-recursive systems have one.
• Infinite impulse response — an impulse response that never quite ends. Recursive systems normally have one.
• Unit circle — the circle of radius one on the plane where the poles of a discrete-time system are marked.
• Delay element — a block that holds a value for one interval.
• Homogeneous solution — the part of the answer set by the initial values.
• Particular solution — the part set by the input.

The mental model

A difference equation relates the output now to a few earlier outputs and a few present and earlier inputs, each with a coefficient. It plays the part the differential equation plays in continuous time, and it is far more pleasant to work with. Computing with it needs nothing but multiplication and addition.

Solving one is direct. Given the coefficients, the initial values and the input, compute the output at the first position, then use that to compute the next, and continue. You can do this on paper, and doing it once for five or six positions is worth more than any amount of reading about it.

The answer separates into two parts, exactly as it does in continuous time. One part is set by the initial values and owes nothing to the input; the other is set by the input. For a stable system the first part dies away, which is why the long-run behaviour depends on the input alone.

Transform the equation, using the transform from the transforms module built for sequences, and the delays become powers. What is left is a ratio of two polynomials again: the transfer function, in the discrete-time setting. Its poles and zeros are marked on a plane, as before.

The stability condition changes shape but not spirit. A causal discrete-time system is stable when every pole lies strictly inside the unit circle. The reason mirrors the continuous case: a pole at some position contributes a term whose size is raised to a power at each step. A distance from the origin less than one shrinks; more than one grows without limit. On the circle it neither grows nor shrinks. The left half-plane became the inside of a circle, and the argument is the same one.

The same warning about the region of convergence applies here too, and matters more than it did. The same ratio of polynomials describes a causal system when the region lies outside the outermost pole, and a non-causal one otherwise. Naming the region is part of naming the system.

Now the distinction that shapes practical work. A system with no feedback path uses input values only. Its impulse response has finite length, because after the last input value passes through, nothing is left circulating. A finite impulse response system cannot be unstable, whatever its coefficients, which is a considerable comfort. It also has a phase that can be made exactly linear, which the previous topics showed is what preserves a waveform's shape.

A system with feedback normally has an impulse response that never quite ends. It can achieve a much sharper response for the same amount of arithmetic, sometimes by a large factor. It can also be unstable, and its coefficients have to be checked. That is a real engineering trade, made deliberately, and both kinds are used constantly.

Any difference equation can be drawn with three elements: a delay, a gain, and an adder. The drawing is close to an implementation, whether that means lines of code or blocks in hardware. Reading a drawing back into an equation is the check that you have understood it. Feedback is visible in such a picture as a path from an output back to an adder. That is what makes the recursive and non-recursive kinds distinguishable on sight.

Finally the frequency response, which closes the module. Evaluate the transfer function on the unit circle and you have the frequency response of the discrete-time system. It is the same object as before: magnitude saying what is kept, phase saying what is delayed.

It has one new feature. It necessarily repeats, because going once around the circle returns to the start. In signal terms, a sampled description cannot distinguish two frequencies separated by the sampling rate. The response to one is the response to the other.

That is the previous topic's repeating spectrum, seen from the other side. The two facts are the same fact, and noticing that is a good place to end. Sampling repeats a spectrum; a discrete-time frequency response repeats for the same reason; and both are consequences of having thrown away everything between the samples.

What you should now be able to explain or do

• Write a difference equation and solve one by stepping forward from the initial values.
• Separate the answer into the part set by the initial values and the part set by the input.
• Obtain the discrete-time transfer function and mark its poles relative to the unit circle.
• State the stability condition and explain why it is the same argument as the continuous-time one.
• Distinguish finite from infinite impulse response systems and state what each buys and costs.
• Draw a difference equation using delays, gains and adders, and read such a drawing back.
• Explain why the frequency response of a discrete-time system repeats, and connect that to sampling.

Check yourself