# What is the event that repeats itself in sinusoidal signals

I am studying medical imaging and it has been stated that it is sufficient for Nyquist Theorem to hold for the signal to be Bandlimited. I am trying to understand what is the wave in a signal.

What is the wave in a signal, a variation in magnitude, of which physical quantity?

For example we have an MRI and we recond the signal is the wave a variation in time of voltage, current?

What is the wave in a signal, a variation in magnitude, of which physical quantity?

Typically in signal processing we don't talk about a "wave" in a signal. In signal processing we just talk about a signal.

A signal is a mathematical abstraction -- for the purposes of "pure" signal processing it's a function of one or more independent variables that is known.

It's up to you to map that from, and back to, the real world.

If you're working on one-dimensional signal processing, like the majority of us here are doing, then you're usually thinking of signals that are a function of time -- so you might be thinking of an audio signal from a microphone, or a radio signal picked up by an antenna, or some process variable in a control system. All of these would be expressed like $$x(t)$$, $$y(t)$$, $$u(t)$$, etc.

For typical 1D signal processing, your signals can be positive or negative, so you don't restrict yourself to thinking about the signal magnitude -- just its value.

If you're working higher dimensional signal processing, such as image processing, then your signal would have more independent variables, i.e. $$u(x, y)$$ might represent the pixel amplitude of a photograph or an x-ray image. I honestly don't know how MRI or CAT scans are stored, but for doing image processing it's probably convenient to represent them as a 3-dimensional amplitude of some sort, i.e. $$u(x, y, z)$$.

If you're thinking in terms of the Nyquist theorem, then in the 2-D case you're assuming a model for your signal that says something like

$$u(x, y) = \sum_n \sum_m k_{n, m}e^{-2j \pi (\frac n N x + \frac m M y)}$$

In this case the "waves", if you want to think in those terms, are the sinusoids generated by each $$k_{n, m}e^{-2j \pi (\frac n N x + \frac m M y)}$$.

Viewing your signal in the Fourier domain, these "waves" are always there -- even if the signal itself has no repeating parts. Mathematically, this is a tremendously powerful feature of the Fourier transform, because it makes it easy to solve linear differential equations (which was Fourier's motivation). Intuitively, this can be a tremendously powerful roadblock to comprehension of the Fourier transform, because those spectral components are there even when a signal isn't repetitive in the least.

If you see "wave" in the literature, or if you're tempted to use it yourself, then "spectral component" is the best general term, and "a component at frequency $$f$$" is probably the best term if you're being specific.

• Thanks Tim! .... Jan 13 at 22:59
• Tim, your last equation didn't make sense to me, so I've edited it to try to resolve my confusion. Please revert or correct if I've got it wrong.
– Peter K.
Jan 13 at 23:10
• @PeterK. Looks much clearer to me -- thanks. Jan 14 at 3:49
• @TimWescott Don't we talk about signal frequency and waveform? Aren't those characteristics of Waves? In the X-axis we most frequently measure time (there is time variation) but what are we measuring in the Y-axis? Jan 14 at 9:19
• "X-axis we most frequently measure time (there is time variation) but what are we measuring in the Y-axis?" Please read my answer fully. The measurement (or hypothesis) in the Y axis is whatever fits the problem at hand. Jan 14 at 17:58

In the context of signal processing, a "wave" in a signal would refer to a variation or fluctuation in some physical quantity over time or space.

In the context of MRI, the wave in the signal is a variation over time of the induced voltage, which results from changes in magnetic fields and the behavior of protons in the body's tissues.

The Nyquist Theorem's relevance in this context is to ensure that the sampling rate of these signals is sufficient to accurately reconstruct the image without losing information.