If you would write the frequency-response in terms of projections through inproducts. How would you do it?

While studying for signals and systems I realised that the most intuitive way of understanding Fourier Series for me, was to see it as a projection (through inproducts) of a signal onto the orthogonal complex exponentials.

Is it possible to view the impulse-response and frequency-response of a signal in this same manner? Looking at the definition it seems clear that there should be such an 'intuitive' understanding, but I am having difficulties seeing it..

• What do you mean by the "impulse-response of a signal"? The frequency response clearly is the inner product of the signal with the complex exponential. – Matt L. Aug 12 '20 at 8:05

the impulse-response and frequency-response of a signal

Only systems have impulse responses and frequency responses, signals don't. I assume that what you mean here.

A System describes the relation ship between its input signal and its output signal. For an LTI system, that relationship can be captured through either the transfer function or the impulse response. The idea is similar

1. Project the signal onto the some orthogonal basis
2. Run the basis function through the system
3. Assemble output basis function into the output signal. Since it's an LTI system you can simply sum or integrate.

Of course that only makes sense if the bases functions are "easier" to run through the system than some arbitrary signal. For the transfer functions the basis functions are complex exponentials and for the impulse response they are time shifted dirac deltas.

Update

Elaborating on step 2: The basis functions have an index or parameter. For example for a complex exponential, the parameter is frequency. Now you calculate the output of the system with the basis function as a an input using the parameter as a variable. Specifically you calculate the output for a complex exponential with frequency $$\omega$$ as a variable. The result is a function of frequency: That's the transfer function.

So if you have a signal $$x(t)$$ that can be expressed as the sum of some basis functions b_k(t) like

$$x(t) = \sum a_k \cdot b_k(t)$$

You can calculate the output as a weighted sum of the outputs of the basis functions.

$$y(t) = T\begin{Bmatrix} x(t) \end{Bmatrix} = T\begin{Bmatrix} \sum a_k \cdot b_k(t) \end{Bmatrix} = \sum a_k \cdot T\begin{Bmatrix} b_k(t) \end{Bmatrix}$$

• Yes, I meant system! Thank you for correcting that! Ok, so this is where I find it difficult to link the concepts. Your 1st point is very clear to me. You can see any n-dimensional signal as composed of n-orthogonal vectors. So FR is just a change of base-vectors. But then in your second step. What is 'running' your signal through the system? Inuitively I would presume it to be a linear transformation (in LTI) in the space with your new base. But how does this work? Does this make the transferft the representation of this TF and thus the impulse response the same returning to B0? – Laurens Aug 13 '20 at 10:41