# Amplitude vs Frequency Response

Consider the simple first-order low pass filter, described by

$\tau \frac{d \mathcal{O}}{dt} + \mathcal{O}(t)=\mathcal{I}(t)$

Considered as a linear system (ie $\mathcal{L} \mathcal{O} = \mathcal{I}$), $\mathcal{L}$ has transfer function $\frac{1}{1 + i \omega \tau}$ (up to maybe some constants, minus signs etc etc depending on convention). By transfer function, I mean $\tilde{R}(k)$ such that if $\tilde{\mathcal{I}}$ denotes the Fourier/Laplace transform of the input, then $\tilde{\mathcal{O}}(t) = \tilde{R}(t)\tilde{\mathcal{I}}(t)$. Questions:

1) Is the transfer function the 'same thing' as the frequency response? The reference textbook I have (Signals and Systems by Oppenheim & Willsky) talks of the frequency response of the 1st order LPF $H(i\omega) = \frac{1}{1 +i \omega \tau}$ which as far as I can tell amounts to something very similar.

2) If we solve the LPF ODE for a sinusoidal input $sin(\omega t)$, then we get an expression for the output amplitude $A(\omega) = \frac{1}{\sqrt{1+\omega^2 \tau^2}}$. How is this related to the transfer function/frequency response?

Supplementary information/question: I'm actually a mathematics student in the situation of having to learn a lot of signals processing very quickly this summer. Are there any textbooks written out there which explain signals and systems in the perspective of a maths student? As far as I can tell, most books seem to pitched towards engineers (and also use $j$ instead of $i$), and are a little different to what I'm used to - I don't want to learn lots of things twice from two different perspectives.

Thank you very much for you time.

• So it turns out that I haven't really thought about this: the amplitude is just the modulus of the transfer function. – Kris Jun 26 '12 at 21:23
• If you could give me some idea of what you need to prepare for in the fall, I might be able to give you some suggestions. – datageist Jun 27 '12 at 2:18
• @DilipSarwate: I think you mislinked; personal budgeting seems a bit off topic for this question. :) – Jason R Jun 27 '12 at 3:02
• @Kris: Signal processing is a rather applied field, so most content that you find is going to be geared toward engineering, thus the use of $j$ for the imaginary unit. I don't think you'll find many radically different perspectives toward the content. As datageist noted, however, if you can tell us more about what you need to learn, we can help point you in the right direction. – Jason R Jun 27 '12 at 3:04
• @JasonR Whoops! See the comment following this answer. – Dilip Sarwate Jun 27 '12 at 11:07