Relationship between Fourier coefficients and power spectral density

In summary, Parseval's Theorem relates the phase ##\phi(x,y)## to the power spectral density ##\Phi(f_{x_n},f_{y_m})## by equating the integral of the squared magnitude of the Fourier coefficients ##|c_{n,m}|^2## to the integral of the power spectral density. In the discrete case, this integral can be approximated as a sum over frequency bins, resulting in the equation ##|c_{n,m}|^2 \approx \Phi(f_{x_n}, f_{y_m})\sum_{f_{x_n}} \sum_{f_{y_m}}\Delta f_x \Delta f_y##. The ensemble average ##\langle|c_{
  • #1
Skaiserollz89
9
0
Homework Statement
Show that ##\langle|c_{n,m}|^2\rangle=\Phi(f_{x_n},f_{y_m})\Delta f_{x_n} \Delta f_{y_m}##, where
##\langle|c_{n,m}|^2\rangle## is the ensemble average of fourier coefficients of a phase screen given by ##\phi(x,y)=\sum\sum c_{n,m}e^{i2\pi(f_{x_n}x+f_{y_m})} ##, and ##\Phi(f_{x_n},f_{y_m})## is the phase power spectral density is the squared-magnitude of the fourier transform of ##\phi(x,y)##.
Relevant Equations
$$\langle|c_{n,m}|^2\rangle=\Phi(f_{x_n},f_{y_m})\Delta f_{x_n} \Delta f_{y_m}$$

$$\phi(x,y)=\sum\sum c_{n,m}e^{i2\pi(f_{x_n}x+f_{y_m}y)} $$

Parseval's Theorem:
$$\int\int_{-\infty}^{+\infty}|\phi(x,y)|^2 \,dx\,dy=\int\int_{-\infty}^{+\infty}\Phi(f_{x_n},f_{y_m}) \,df_x\,df_y.$$
Here, ##\Phi(f_{x_n},f_{y_m})=|\mathscr{F(\phi(x,y))}|^2 ## is the Power Spectral Density of ##\phi(x,y)## and ##\mathscr{F}## is the Fourier transform operator.

Parseval's Theorem relates the phase ##\phi(x,y)## to the power spectral density ##\Phi(f_{x_n},f_{y_m})## by

$$\int\int_{-\infty}^{+\infty}|\phi(x,y)|^2 \,dx\,dy=\int\int_{-\infty}^{+\infty}\Phi(f_{x_n},f_{y_m}) \,df_x\,df_y$$

Substitution of ##\phi(x,y)=\sum\sum c_{n,m}e^{i2\pi(f_{x_n}x+f_{y_m}y)} ## into the left side of Parsevals Theorem yields

$$|c_{n,m}|^2=\int\int_{-\infty}^{+\infty}\Phi(f_{x_n},f_{y_m}) \,df_x\,df_y$$.In ##\langle|c_{n,m}|^2\rangle=\Phi(f_{x_n},f_{y_m})\Delta f_{x_n} \Delta f_{y_m}##, ##\langle|c_{n,m}|^2\rangle## represents the ensemble average of ##|c_{n,m}|^2## and ##\Delta f_{x_n}## and ## \Delta f_{y_m}## represent the frequency bin widths, so we need to consider the discrete nature of the Fourier transform I think.

In the continuous case, the expression ##|c_{n,m}|^2## represents the contribution of the continuous frequency range within the integral to the squared magnitude of the Fourier coefficient ##c_{n,m}##. However, in the discrete case, we can approximate this integral as a sum over the frequency bins.

Thus, we can rewrite the equation as:

$$ |c_{n,m}|^2 \approx \sum_{f_{x_n}} \sum_{f_{y_m}} \Phi(f_{x_n}, f_{y_m}) \Delta f_x \Delta f_y$$where ##\Delta f_{x_n}## and ##\Delta f_{y_m}## represent the width of the frequency bins.

By approximating ##\Phi(f_{x_n}, f_{y_m})## as constant within each sum, it can be moved out in front of the double sum. However, it's important to note that this is an approximation and assumes that ##\Phi(f_{x_n}, f_{y_m})## doesn't vary significantly within each sum.

$$ |c_{n,m}|^2 \approx \Phi(f_{x_n}, f_{y_m})\sum_{f_{x_n}} \sum_{f_{y_m}}\Delta f_x \Delta f_y$$From here, I am not really sure what additional simplifications I can make to get to the result that

$$ \langle|c_{n,m}|^2\rangle=\Phi(f_{x_n},f_{y_m})\Delta f_{x_n} \Delta f_{y_m} $$

but it seems as though the ensemble average has something to do with collapsing the summation.
Any additional help here, or suggestions would be much appreciated.
 
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  • #2


One way to approach this problem is by considering the concept of ergodicity. In signal processing, ergodicity refers to the idea that the statistical properties of a signal can be determined by a single realization of the signal, as long as the signal is stationary (its statistical properties do not change over time).

In this case, we can consider our signal to be the function ##\phi(x,y)## and its Fourier transform ##\mathscr{F}[\phi(x,y)]##. By assuming ergodicity, we can say that the ensemble average of ##|c_{n,m}|^2## is equivalent to the time average of the squared magnitude of the Fourier transform at a specific point in space.

Using this idea, we can rewrite the ensemble average as:

$$\langle|c_{n,m}|^2\rangle = \frac{1}{T} \int_0^T |\mathscr{F}[\phi(x,y)]|^2 dt$$

where T represents the total time duration of the signal.

We can then apply Parseval's Theorem to this time average, which gives us:

$$\frac{1}{T} \int_0^T |\mathscr{F}[\phi(x,y)]|^2 dt = \frac{1}{T} \int_0^T \int\int_{-\infty}^{+\infty} \Phi(f_{x_n},f_{y_m}) df_x df_y dt$$

Since ##\Phi(f_{x_n},f_{y_m})## is constant with respect to time, we can move it out of the time integral, giving us:

$$\frac{1}{T} \int_0^T \int\int_{-\infty}^{+\infty} \Phi(f_{x_n},f_{y_m}) df_x df_y dt = \int\int_{-\infty}^{+\infty} \Phi(f_{x_n},f_{y_m}) df_x df_y$$

Now, we can equate this result with the previous expression for the ensemble average, giving us:

$$\int\int_{-\infty}^{+\infty} \Phi(f_{x_n},f_{y_m}) df_x df_y = \langle|c_{n,m}|^2\rangle$$

Comparing this with the previous equation we derived, we can see
 

Related to Relationship between Fourier coefficients and power spectral density

What is the relationship between Fourier coefficients and power spectral density?

The relationship between Fourier coefficients and power spectral density (PSD) lies in the fact that the PSD represents the distribution of power into frequency components, which can be derived from the Fourier coefficients. Specifically, the PSD is proportional to the squared magnitude of the Fourier coefficients of a signal.

How do Fourier coefficients help in calculating power spectral density?

Fourier coefficients represent the amplitude and phase of sinusoidal components of a signal at different frequencies. The power spectral density is calculated by taking the squared magnitude of these Fourier coefficients, which gives the power present at each frequency. This process is often referred to as the periodogram method.

Why is power spectral density important in signal processing?

Power spectral density is crucial in signal processing because it provides a detailed view of how the power of a signal is distributed across different frequencies. This information is essential for analyzing the frequency characteristics of signals, identifying dominant frequencies, and filtering out noise.

Can power spectral density be used to reconstruct the original signal?

No, the power spectral density alone cannot be used to reconstruct the original signal because it only provides information about the power at each frequency, not the phase information. Both magnitude and phase information from the Fourier coefficients are required to accurately reconstruct the original signal.

What are common methods to estimate power spectral density from Fourier coefficients?

Common methods to estimate power spectral density from Fourier coefficients include the periodogram, Welch's method, and the Blackman-Tukey method. These methods involve calculating the squared magnitude of the Fourier coefficients and applying various techniques to improve the accuracy and reduce the variance of the PSD estimate.

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