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I'm currently taking a course on signals and systems, and even though we are much ahead of frequency response, I still feel like I am missing an intuitive grasp of what the frequency response of a signal is. Specifically, when I look at the various CTFT pairs, it's difficult for me to intuitively understand why the pairs make sense.

If someone could provide an intuitive explanation of what frequency response is, that would be much appreciated!

Thanks

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    $\begingroup$ I would use a different terminology here. I think what you mean is the frequency domain representation of signals, or their Fourier transform. In my opinion, only systems have a frequency response (namely the Fourier transform of their impulse response). $\endgroup$ – Deve Dec 2 '14 at 8:37
  • $\begingroup$ Also have a look at this question and its answers. $\endgroup$ – Matt L. Dec 2 '14 at 8:45
  • $\begingroup$ It would help if you gave some examples of Fourier transform pairs that seem counter-intuitive to you. $\endgroup$ – Matt L. Dec 2 '14 at 10:36
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    $\begingroup$ agree with @Deve regarding terminology. systems (particularly linear time-invariant LTI systems) or filters have a "frequency response". but signals have a "spectrum". the Fourier Transform of a signal is its spectrum. the Fourier Transform of the impulse response of a filter is the spectrum of the impulse response and is also the frequency response of the filter. $\endgroup$ – robert bristow-johnson Dec 29 '15 at 0:17
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The frequency response tells you how a system reacts to a sinusoidal excitation of a given frequency, in terms of amplitude and phase.

As any signal, periodic or not, can be described as a sum of sinusoids, this allows you to deduce the response to any signal, by the superposition principle.

Interesting features of the frequency response are the continuous gain, the presence of resonances and the bandwidth. A flat curve reproduces the original excitation without deformations.

You get a feeling of a frequency response when looking at how a slowing spin-drier vibrates.

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Frequency response is the Fourier transform (or DFT in discrete case) of the impulse response.

I think you may be confusing frequency response with frequency domain representation of a signal. Frequency domain representation is simply any time domain signal converted into frequency domain. This representation will tell you the amplitudes of the complex exponentials (or perhaps more simply sinusoids with phase), which the signal consists of.

For the intuition of pairs... well in some cases there might be a lot of intuition to them, at least when considering all of the Fourier transform's properties. In others cases it may be more of a mathematical exercise. Just plug a function to the formulas and see if both the Fourier transform and it's inverse return similar functions (remember to convert the rectangular format to polar format in frequency domain, as you are primarily concerned with the magnitudes).

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Frequency interpretation remains difficult, even after some years. Each point in the Fourier spectrum somehow represents a quantity (amplitude on the $y$-axis) of a sine/cosine at a certain frequency (location on the $x$-axis of frequencies). The signal somehow contains a sine with this quantity, and all other sines (with their respective quantities) too.

To get intuition, you can play with different signals, vary their parameters, and see how their Fourier transform change. For instance:

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There is a very intuitive explanation of Fourier representation of signals.

http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

If you mean the frequency representation when you mentioned frequency response this might help you.

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