[Fourier transform and its application study notes] 17. ш function

Three properties of the ш function

Last lesson we learned the $ш_p$ function, which is defined as follows

$ш_p = \displaystyle{\sum_{k=-\infty}^{\infty}\delta (X-KP)}$

The $ш_p$ function has the following three properties,

1) Sampling property, inheriting the sampling property of the $\delta$ function

$f (x) ш_p (x) = \displaystyle{\sum_{k=-\infty}^{\infty}f (KP) \delta (X-KP)}$

2) Periodic properties, inheriting the shift properties of the $\delta$ function

$ (f*ш_p) (x) = \displaystyle{\sum_{k=-\infty}^{\infty}f (X-KP)}$

3) Fourier transform

$\mathcal{f}ш_p = \frac{1}{p}ш_{\frac{1}{p}}$

$\mathcal{f}^{-1}ш_p = \frac{1}{p}ш_{\frac{1}{p}}$

These three properties of the $ш_p$ function are the basis for derivation later in this lesson.

Interpolation issues

The interpolation problem is our next problem, we need to use a reliable method to interpolate discrete measurements or sample values, through interpolation, we can get all the values of the sampled signal. (The problem here is and "what we ' re actually gonna solve in a quite remarkable" is the exact interplation of value of a function from a discrete set of measurement or a discrete set of samples. We ll be able to interpolate all values of a signal or a function from a discrete set of samples.)

Suppose there is a process that changes over time, taking measurements of the process at equal intervals (such as fractions of a second), getting a set of measurement data $ (T_0,Y_0), (T_1,y_1), (t_2,y_2) $ and so on, we can use this data as a series of points

We can fit a curve to the sampled data (fit a curve to the data) or interpolate in the midpoint based on the measured data (interpolate values of the process at the intermediate points bas Ed on the measurements).

Curve Fitting and interpolation

1) curve fitting is a curve function based on sampled data, and the value on the function is the approximate value of the original signal.

2) interpolation is through the generation of the formula, to find out the number of points in the middle of the two sampling point. Interpolation is not necessarily linear, and the specific values are determined according to the interpolation formula.

Although the methods are different, the aim is to get the original signal from the limited sampling point.

A point value other than the sample point is required, there is no fixed method, and the appropriate method needs to be chosen according to the actual situation. Of course, more measured values can provide higher curve fitting or more accurate interpolation.

Uncertainty between sampling points

The uncertainty of interpolation and fitting, from an extreme point of view, can be seen as oscillation, that is, how quickly the function changes from one point to another. The more frequently a function turns, the greater the uncertainty of curve fitting or interpolation, and we need to understand and control (regulate) this uncertainty.

We analyze the signal from two aspects in the time domain and frequency domain, and the Fourier transform in the frequency domain reflects the frequency component of the signal, so we can analyze the speed of the function oscillation. The high frequency of Fourier transform is correlated with fast oscillation, and the low frequency of Fourier transform is correlated with low speed oscillation. To understand the oscillation speed of the signal, we need to analyze its Fourier transform.

The way to resolve this uncertainty is to specify the maximum frequency at which the function allows oscillation. If we remove the high frequency after the Fourier transform, it is equivalent to removing the fast oscillation.

The workaround can be summarized in the following definition:

For a function F (x) with a finite bandwidth, if its Fourier transform has a constant value of zero in a band other than the Fourier transform $\mathcal{f}f (s) \equiv 0 \ for \ |s|\geqslant \frac{p}{2}$, the minimum $p$ value is called bandwidth.

For the limited bandwidth signal, the uncertainty problem can be completely solved, that is, the function expression that can be worth to the signal according to the discrete sampling $f (x) $

The derivation process is as follows:

The periodicity of $\mathcal{f}f (s) $ by $ш_p$ using the cyclic nature of the $ш_p$

$\mathcal{f}f *ш_p$

Recovering from the periodic Fourier transform to the original Fourier transform

$\mathcal{f}f = \pi_p (\mathcal{f}f *ш_p) $

Then the Fourier inverse transformation is obtained to obtain the time domain signal.

$\begin{align*}
F (t)
&=\mathcal{f}^{-1} (\pi_p (\mathcal{f}f*ш_p)) \ \
&= (\mathcal{f}^{-1}\pi_p) * (\mathcal{f}^{-1} (\mathcal{f}f*ш_p)) \qquad (fourier\ convolution\ Thereom) \ \
&= (Psinc (PT)) * ((\mathcal{f}^{-1}\mathcal{f}f (t)) (\mathcal{f}^{-1}ш_p (t))) \ \
&= (Psinc (PT)) * (f (t) \cdot \frac{1}{p}ш_{\frac{1}{p}} (t)) \qquad (fourier\ transform\ of\ш_p) \ \
&= (Psinc (PT)) * (\frac{1}{p}f (t) \sum_{k=-\infty}^{\infty}\delta (x-\frac{k}{p)) \ \
&= (Psinc (PT)) * (\frac{1}{p}\sum_{k=-\infty}^{\infty}f (\frac{k}{p}) \delta (x-\frac{k}{p)) \qquad (Ш_p\ Sampling \ property) \ \
&=\sum_{k=-\infty}^{\infty}f (\frac{k}{p}) sinc (PT) *\delta (x-\frac{k}{p}) \ \
&=\sum_{k=-\infty}^{\infty}f (\frac{k}{p}) sinc (P (t-\frac{k}{p})) \qquad \delta\ Shift\ Property
\end{align*}$

Therefore, for the finite bandwidth function $f (t) $ can be written as follows,

$f (t) = \displaystyle{\sum_{k=-\infty}^{\infty}f (\frac{k}{p}) sinc (P (t-\frac{k}{p}))}\quad, \quad \mathcal{f}f (s) \ EQUIV 0 \ for\ |s|\geqslant\frac{p}{2}$

The conclusion is:

• For a function $f (t) $ with a bandwidth of $p$, if the sampling interval is $\frac{1}{p}$ and all sample points are known to $\displaystyle{\sum_{k=-\infty}^{\infty}f (\frac{k}{p}), the value of}$ Then we can get all the values of the original function $f (t) $ by inserting the formula.

This is called the sampling theorem (sampling theorem), which can be said to be the most important formula for the entire course.

[Fourier transform and its application study notes] 17. ш function