Signals and Systems - Alternate notation for systems of signals?

Question

I specifically have a problem with certain notation used to describe systems of signals (context: Signals and Systems by Oppenheim, et. al.); from the Center for Computer Research in Music and Acoustics (Stanford University):

The current material I am learning with likes to use something similar along these lines. Particularly, in proofs, I have a hard time differentiating between the system proper and the inputs to the system, as the system is not allocated a specific variable to represent it before evaluation/substitution/abstraction or "binding."

Can I represent a system with a more function-like notation: e.g.:

$$H[f(x)] = f(2x)$$

To me, this seems a lot clearer, but it seems to break down in the context of ordinary differential equations:

$$\frac{dy(t)}{dt} + (sin\ t)(y(t)) = 3x(t)$$

Would the following be a good notation for the above?

$$H[y(t), x(t)] = \{ \frac{dy(t)}{dt} + (sin\ t)(y(t)) = 3x(t) \}$$

Or, am I misunderstanding something that is causing me to "not get" the notation?

I am looking for a clear notation that expresses the signals and their dependencies clearly.

Answer 1

I am not sure what you exactly mean.

In general, $x[n]$ represents the input of the system, and y[n] the output. (Except for the specific illustration)

We always use $H(\cdot)$ to represent a filter. ( But you can use it to represent a general transform , just the notation)

If you define the function $H(\cdot)$ with the first argument $x[n]$ and the second one $y[n]$, then it seems no problem.

An advise: if you want to ask the questions about Signal Processing, please post question on this page :)

Answer 2

The problem is that in practice a linear systems is not "naturally" given by an explicit function $H(x) = y$ that maps an input signal $x$ to an output signal $y$, but rather an implicit relation like your example: $$ y'(t) + \sin t \, y(t) = 3 x(t), \quad y(0) = y_0 \quad (1) $$ Provided this equation has a unique solution, this equation represents a linear system $H$ that maps an input signal $x$ to an output $y$, but we have no explicit formula for $H$ in this case.

To answer your question, I think the best notation is to stick with (1), and simply state which signal is the input and which signal is the output if you need to be extra clear.