Power-Spectral Density

Wiener-Khintchine Theorem

In the remainder of this paper we restrict ourselves to ergodic processes. As was shown in the last chapter the following statements then hold:

Each individual pattern function $x_i(t)$ is representative of the entire random process $\{x_i(t)\}$.
All time means are thus identical to the corresponding coulter means.
The autocorrelation function, which is generally affected by the two time parameters $t_1$ and $t_2$, now depends only on the time difference $τ = t_2 - t_1$:

$$\varphi_x(t_1,t_2)={\rm E}\big[x(t_{\rm 1})\cdot x(t_{\rm 2})\big] = \varphi_x(\tau)= \int^{+\infty}_{-\infty}x(t)\cdot x(t+\tau)\,{\rm d}t.$$

The auto-correlation function provides quantitative information about the (linear) statistical bindings within the ergodic process $\{x_i(t)\}$ in the time domain. The equivalent descriptor in the frequency domain is the "power-spectral density", often also referred to as the "power density spectrum".

$\text{Definition:}$ The power-spectral density $\rm (PSD)$ of an ergodic random process $\{x_i(t)\}$ is the Fourier transform of the auto-correlation function $\rm (ACF)$:

$${\it \Phi}_x(f)=\int^{+\infty}_{-\infty}\varphi_x(\tau) \cdot {\rm e}^{- {\rm j\hspace{0.05cm}\cdot \hspace{0.05cm} \pi}\hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}\tau} {\rm d} \tau. $$

This Functional relationship is called the "Theorem of Wiener and Khinchin".

Similarly, the auto-correlation function can be computed as the inverse Fourier transform of the power-spectral density (see section "Inverse Fourier transform" in the book "Signal Representation"):

$$ \varphi_x(\tau)=\int^{+\infty}_{-\infty} {\it \Phi}_x \cdot {\rm e}^{- {\rm j\hspace{0.05cm}\cdot \hspace{0.05cm} \pi}\hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}\tau} {\rm d} f.$$

The two equations are directly applicable only if the random process contains neither a DC component nor periodic components.
Otherwise, one must proceed according to the specifications given in section "Power-spectral density with DC component".

Physical interpretation and measurement

The lower chart shows an arrangement for (approximate) metrological determination of the power-spectral density ${\it \Phi}_x(f)$. The following should be noted in this regard:

The random signal $x(t)$ is applied to a (preferably) rectangular and (preferably) narrowband filter with center frequency $f$ and bandwidth $Δf$ where $Δf$ must be chosen sufficiently small according to the desired frequency resolution.
The corresponding output signal $x_f(t)$ is squared and then the mean value is formed over a sufficiently long measurement period $T_{\rm M}$. This gives the "power of $x_f(t)$" or the "power components of $x(t)$ in the spectral range from $f - Δf/2$ to $f + Δf/2$":

To measure the power-spectral density

$$P_{x_f} =\overline{x_f(t)^2}=\frac{1}{T_{\rm M}}\cdot\int^{T_{\rm M}}_{0}x_f^2(t) \hspace{0.1cm}\rm d \it t.$$

Division by $Δf$ leads to the power-spectral density $\rm (PSD)$:

$${{\it \Phi}_{x \rm +}}(f) =\frac{P_{x_f}}{{\rm \Delta} f} \hspace {0.5cm} \Rightarrow \hspace {0.5cm} {\it \Phi}_{x}(f) = \frac{P_{x_f}}{{\rm 2 \cdot \Delta} f}.$$

${\it \Phi}_{x+}(f) = 2 \cdot {\it \Phi}_x(f)$ denotes the one-sided PSD defined only for positive frequencies. For $f<0$ ⇒ ${\it \Phi}_{x+}(f) = 0$. In contrast, for the commonly used two-sided power-spectral density:

$${\it \Phi}_x(-f) = {\it \Phi}_x(f).$$

While the power $P_{x_f}$ tends to zero as the bandwidth $Δf$ becomes smaller, the power-spectral density remains nearly constant above a sufficiently small value of $Δf$. For the exact determination of ${\it \Phi}_x(f)$ two boundary crossings are necessary:

$${{\it \Phi}_x(f)} = \lim_{{\rm \Delta}f\to 0} \hspace{0.2cm} \lim_{T_{\rm M}\to\infty}\hspace{0.2cm} \frac{1}{{\rm 2 \cdot \Delta}f\cdot T_{\rm M}}\cdot\int^{T_{\rm M}}_{0}x_f^2(t) \hspace{0.1cm} \rm d \it t.$$

$\text{Conclusion:}$

From this physical interpretation it further follows that the power-spectral density is always real and can never become negative.
The total power of the random signal $x(t)$ is then obtained by integration over all spectral components:

$$P_x = \int^{\infty}_{0}{\it \Phi}_{x \rm +}(f) \hspace{0.1cm}{\rm d} f = \int^{+\infty}_{-\infty}{\it \Phi}_x(f)\hspace{0.1cm} {\rm d} f .$$

Reciprocity law of ACF duration and PSD bandwidth

All the Fourier transform theorems derived in the book "Signal Representation" for deterministic signals can also be applied to

the auto-correlation function $\rm (ACF)$, and
the power-spectral Density $\rm (PSD)$.

On the "Reciprocity Law" of ACF and PSD

However, not all laws yield meaningful results due to the specific properties

of auto-correlation function (always real and even)
and power-spectral density (always real, even, and non–negative).

We now consider as in the section Interpretation of the auto-correlation function two different ergodic random processes $\{x_i(t)\}$ and $\{y_i(t)\}$ based on

two pattern signals $x(t)$ and $y(t)$ ⇒ upper sketch,
two auto-correlation functions $φ_x(τ)$ and $φ_y(τ)$ ⇒ middle sketch,
two power-spectral densities ${\it \Phi}_x(f)$ and ${\it \Phi}_y(f)$ ⇒ bottom sketch.

Based on these graphs, the following statements can be made:

The areas under the PSD curves are equal ⇒ the processes $\{x_i(t)\}$ and $\{y_i(t)\}$ have equal power:

$${\varphi_x({\rm 0})}\hspace{0.05cm} =\hspace{0.05cm} \int^{+\infty}_{-\infty}{{\it \Phi}_x(f)} \hspace{0.1cm} {\rm d} f \hspace{0.2cm} = \hspace{0.2cm}{\varphi_y({\rm 0})} = \int^{+\infty}_{-\infty}{{\it \Phi}_y(f)} \hspace{0.1cm} {\rm d} f .$$

The from classical (deterministic) system theory well known "Reciprocity Theorem of time duration and bandwidth" also applies here: A narrow ACF corresponds to a broad PDS and vice versa.
As a descriptive quantity, we use here the equivalent PSD bandwidth $∇f$ $($one speaks "Nabla-f"$)$,
similarly defined as the equivalent ACF duration $∇τ$ in chapter "Interpretation of the auto-correlation function":

$${{\rm \nabla} f_x} = \frac {1}{{\it \Phi}_x(f = {\rm 0})} \cdot \int^{+\infty}_{-\infty}{{\it \Phi}_x(f)} \hspace{0.1cm} {\rm d} f, $$

$${ {\rm \nabla} \tau_x} = \frac {\rm 1}{ \varphi_x(\tau = \rm 0)} \cdot \int^{+\infty}_{-\infty}{\varphi_x(\tau )} \hspace{0.1cm} {\rm d} \tau.$$

With these definitions, the following basic relationship holds:

$${{\rm \nabla} \tau_x} \cdot {{\rm \nabla} f_x} = 1\hspace{1cm}{\rm resp.}\hspace{1cm} {{\rm \nabla} \tau_y} \cdot {{\rm \nabla} f_y} = 1.$$

$\text{Example 1:}$ We start from the graph at the top of this page:

The characteristics of the higher frequency signal $x(t)$ are $∇τ_x = 0.33\hspace{0.08cm} \rm µs$ and $∇f_x = 3 \hspace{0.08cm} \rm MHz$.
The equivalent ACF duration of the signal $y(t)$ is three times: $∇τ_y = 1 \hspace{0.08cm} \rm µs$.
The equivalent PSD bandwidth of the signal $y(t)$ is thus only $∇f_y = ∇f_x/3 = 1 \hspace{0.08cm} \rm MHz$.

$\text{General:}$ The product of equivalent ACF duration ${ {\rm \nabla} \tau_x}$ and equivalent PSD bandwidth $ { {\rm \nabla} f_x}$ is always "one":

$${ {\rm \nabla} \tau_x} \cdot { {\rm \nabla} f_x} = 1.$$

$\text{Proof:}$ According to the above definitions:

$${ {\rm \nabla} \tau_x} = \frac {\rm 1}{ \varphi_x(\tau = \rm 0)} \cdot \int^{+\infty}_{-\infty}{ \varphi_x(\tau )} \hspace{0.1cm} {\rm d} \tau = \frac { {\it \Phi}_x(f = {\rm 0)} }{ \varphi_x(\tau = \rm 0)},$$

$${ {\rm \nabla} f_x} = \frac {1}{ {\it \Phi}_x(f = {\rm0})} \cdot \int^{+\infty}_{-\infty}{ {\it \Phi}_x(f)} \hspace{0.1cm} {\rm d} f = \frac {\varphi_x(\tau = {\rm 0)} }{ {\it \Phi}_x(f = \rm 0)}.$$

Thus, the product is equal to $1$.

q.e.d.

$\text{Example 2:}$ A limiting case of the reciprocity law represents the so-called "White Noise":

This includes all spectral components (up to infinity).
The equivalent PSD bandwidth $∇f$ is infinite.

The law given here states that for the equivalent ACF duration $∇τ = 0$ must hold ⇒ white noise has a Dirac-shaped ACF.

For more on this topic, see the three-part (German language) learning video The AWGN channel, especially the second part.

Power-spectral density with DC component

We assume a DC–free random process $\{x_i(t)\}$. Further, we assume that the process also contains no periodic components. Then holds:

The auto-correlation function $φ_x(τ)$ vanishes for $τ → ∞$.
The power-spectral density ${\it \Phi}_x(f)$ – computable as the Fourier transform of $φ_x(τ)$ – is both continuous in value and continuous in time, i.e., without discrete components.

We now consider a second random process $\{y_i(t)\}$, which differs from the process $\{x_i(t)\}$ only by an additional DC component $m_y$:

$$\left\{ y_i (t) \right\} = \left\{ x_i (t) + m_y \right\}.$$

The statistical descriptors of the mean-valued random process $\{y_i(t)\}$ then have the following properties:

The limit of the ACF for $τ → ∞$ is now no longer zero, but $m_y^2$. Throughout the $τ$–range from $-∞$ to $+∞$ the ACF $φ_y(τ)$ is larger than $φ_x(τ)$ by $m_y^2$:

$${\varphi_y ( \tau)} = {\varphi_x ( \tau)} + m_y^2 . $$

According to the elementary laws of the Fourier transform, the constant ACF contribution in the PSD leads to a Dirac delta function $δ(f)$ with weight $m_y^2$:

$${{\it \Phi}_y ( f)} = {\Phi_x ( f)} + m_y^2 \cdot \delta (f). $$

More information about the $\delta$–function can be found in the chapter Direct current signal - Limit case of a periodic signal of the book "Signal Representation". Furthermore, we would like to refer you here to the (German language) learning video "Herleitung und Visualisierung der Diracfunktion" ⇒ "Derivation and visualization of the Dirac delta function".

Numerical PSD determination

Auto-correlation function and power-spectral density are strictly related via the Fourier transform. This relationship also holds for discrete-time ACF representation with the sampling operator ${\rm A} \{ \varphi_x ( \tau ) \} $, thus for

$${\rm A} \{ \varphi_x ( \tau ) \} = \varphi_x ( \tau ) \cdot \sum_{k= - \infty}^{\infty} T_{\rm A} \cdot \delta ( \tau - k \cdot T_{\rm A}).$$

The transition from the time domain to the spectral domain can be derived with the following steps:

The distance $T_{\rm A}$ of two samples is determined by the absolute bandwidth $B_x$ (maximum occurring frequency within the process) via the sampling theorem:

$$T_{\rm A}\le\frac{1}{2B_x}.$$

The Fourier transform of the discrete-time (sampled) ACF yields an PSD periodic with ${\rm 1}/T_{\rm A}$ :

$${\rm A} \{ \varphi_x ( \tau ) \} \hspace{0.3cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.3cm} {\rm P} \{{{\it \Phi}_x} ( f) \} = \sum_{\mu = - \infty}^{\infty} {{\it \Phi}_x} ( f - \frac {\mu}{T_{\rm A}}).$$

$\text{Conclusion:}$ Since both $φ_x(τ)$ and ${\it \Phi}_x(f)$ are even and real functions, the relation holds:

$${\rm P} \{ { {\it \Phi}_x} ( f) \} = T_{\rm A} \cdot \varphi_x ( k = 0) +2 T_{\rm A} \cdot \sum_{k = 1}^{\infty} \varphi_x ( k T_{\rm A}) \cdot {\rm cos}(2{\rm \pi} f k T_{\rm A}).$$

The power density spectrum (PSD) of the continuous-time process is obtained from ${\rm P} \{ { {\it \Phi}_x} ( f) \}$ by bandlimiting to the range $\vert f \vert ≤ 1/(2T_{\rm A})$.
In the time domain, this operation means interpolating the individual ACF samples with the ${\rm si}$–function, where ${\rm si}(x)$ stands for $\sin(x)/x$ .

Discrete-time ACF and periodically continued PSD

$\text{Example 3:}$ A Gaussian ACF $φ_x(τ)$ is sampled at distance $T_{\rm A}$ where the sampling theorem is satisfied:

The Fourier transform of the discrete-time ACF ${\rm A} \{φ_x(τ) \}$ be ${\rm P} \{ { {\it \Phi}_x} ( f) \}$.
This with ${\rm 1}/T_{\rm A}$ periodic function ${\rm P} \{ { {\it \Phi}_x} ( f) \}$ is accordingly infinitely extended ( red curve ).
The PSD ${\it \Phi}_x(f)$ of the continuous-time process $\{x_i(t)\}$ is obtained by band-limiting to the frequency range highlighted in blue in the figure $\vert f - T_{\rm A} \vert ≤ 0.5$.

Accuracy of the numerical PSD calculation

For the following analysis, we make the following assumptions:

The discrete-time ACF $φ_x(k - T_{\rm A})$ was determined numerically from $N$ samples. As already shown on Accuracy of the numerical ACF calculation these values are in error and the errors are correlated if $N$ was chosen too small.
To calculate the periodic power density spectrum (PSD), we use only the ACF values $φ_x(0)$, ... , $φ_x(K - T_{\rm A})$:

$${\rm P} \{{{\it \Phi}_x} ( f) \} = T_{\rm A} \cdot \varphi_x ( k = 0) +2 T_{\rm A} \cdot \sum_{k = 1}^{K} \varphi_x ( k T_{\rm A})\cdot {\rm cos}(2{\rm \pi} f k T_{\rm A}).$$

$\text{Conclusion:}$ The accuracy of the PSD calculation is determined to a strong extent by the parameter $K$ :

If $K$ is chosen too small, the ACF values actually present $φ_x(k - T_{\rm A})$ with $k > K$ will not be taken into account.
If $K$ is too large, also such ACF values are considered, which should actually be zero and are finite only because of the numerical ACF calculation.
These values, however, are - due to a too small $N$ in the ACF calculation - only errors, and impair the PSD calculation more than they provide a useful contribution to the result.

Accuracy of numerical PSD calculation

$\text{Example 4:}$ We consider here a zero mean process with statistically independent samples.

Thus, only the ACF value $φ_x(0) = σ_x^2$ should be different from zero.
But if one determines the ACF numerically from only $N = 1000$ samples, one obtains finite ACF values even for $k ≠ 0$ .

The upper figure shows that these erroneous ACF–values can be up to $6\%$ of the maximum value.
The numerically determined power density spectrum is shown below. The theoretical curve is shown in yellow, which was calculated for $\vert f - T_{\rm A} \vert ≤ 0.5$ should be constant.
The green and purple curves illustrate how by $K = 3$ and $K = 10$ respectively, the result is distorted compared to $K = 0$ .

In this case (statistically independent random variables) the error grows monotonically with increasing $K$. In contrast, for a random variable with statistical bindings, there is an optimal value for $K$ in each case.

If this is chosen too small, significant bindings are not considered.
In contrast, a value that is too large leads to oscillations that can only be attributed to erroneous ACF values

.