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What is Energy Spectral Density?
Energy Spectral Density
The distribution of the energy of a signal in the frequency domain is known as energy spectral density (ESD) or energy density (ED) or energy density spectrum. The ESD function is denoted by $\mathrm{\mathit{\psi \left ( \omega \right )}}$ and is given by,
$$\mathrm{\mathit{\psi \left ( \omega \right )\mathrm{=}\left|X\left ( \omega \right ) \right|^{\mathrm{2}}}}$$
For an energy signal, the total area under the energy spectral density curve plotted as the function of frequency is equal to the total energy of the signal.
Explanation
Consider a linear system having $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ and $\mathrm{\mathit{y\left ( \mathit{t} \right )}}$ as input and output respectively. Then, the Fourier transform of $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ and $\mathrm{\mathit{y\left ( \mathit{t} \right )}}$ be
$$\mathrm{\mathit{x\left ( t \right )\overset{FT}{\leftrightarrow}X\left ( \omega \right )}}$$
$$\mathrm{\mathit{y\left ( t \right )\overset{FT}{\leftrightarrow}Y\left ( \omega \right )}}$$
And the transfer function of the system is π»(π). Then, we get,
$$\mathrm{\mathit{Y\left ( \omega \right )\mathrm{=}H\left ( \omega \right )\cdot X\left ( \omega \right )\; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$
Therefore, the ESD of the input and output signals is given by,
ESD of input function, $\mathrm{\mathit{\psi_{x} \left ( \omega \right )\mathrm{=}\left|X\left ( \omega \right ) \right|^{\mathrm{2}}\; \; \cdot \cdot \cdot \left ( \mathrm{2} \right )}}$
ESD of output function, $\mathrm{\mathit{\psi_{y} \left ( \omega \right )\mathrm{=}\left|Y\left ( \omega \right ) \right|^{\mathrm{2}}\; \; \cdot \cdot \cdot \left ( \mathrm{3} \right )}}$
From equations (1), (2) & (3), we have,
$$\mathrm{\mathit{\psi_{y} \left ( \omega \right )\mathrm{=}\left|Y\left ( \omega \right ) \right|^{\mathrm{2}}\mathrm{=}\left|H\left ( \omega \right )\cdot X\left ( \omega \right ) \right|^{\mathrm{2}}}}$$
$$\mathrm{\mathit{\Rightarrow \psi_{y} \left ( \omega \right )\mathrm{=}\left|H\left ( \omega \right )\right|^{\mathrm{2}}\cdot \left|X\left ( \omega \right ) \right|^{\mathrm{2}}\mathrm{=}\left| H\left ( \omega \right )\right|^{\mathrm{2}}\cdot \psi _{x}\left ( \omega \right )}}$$
$$\mathrm{\mathit{\therefore \psi_{y} \left ( \omega \right )\mathrm{=}\left| H\left ( \omega \right )\right|^{\mathrm{2}} \psi _{x}\left ( \omega \right )\; \; \cdot \cdot \cdot \left ( \mathrm{4} \right )}}$$
Hence, it is clear from eq. (4) that the ESD of the output function of a linear system is the product of the square of the magnitude of the system transfer function and the ESD of the input signal.
Now, the energy of the output signal is,
$$\mathrm{\mathit{E_{y}\mathrm{=}\int_{-\infty }^{\infty }\psi _{y}\left ( f \right )df\; \; \cdot \cdot \cdot \left ( \mathrm{5} \right ) }}$$
$$\mathrm{\mathit{\Rightarrow E_{y}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }\psi _{y}\left ( \omega \right )d\omega\; \; \; \left ( \because f\mathrm{=} \frac{\omega }{\mathrm{2}\pi } \right )}}$$
$$\mathrm{\mathit{\Rightarrow E_{y}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }\left|H\left ( \omega \right ) \right|^{\mathrm{2}}\psi _{x}\left ( \omega \right )d\omega\mathrm{=}\frac{\mathrm{2}}{\mathrm{2}\pi }\int_{\mathrm{0} }^{\infty }\left|H\left ( \omega \right ) \right|^{\mathrm{2}}\psi _{x}\left ( \omega \right )d\omega }}$$
$$\mathrm{\mathit{\Rightarrow E_{y}\mathrm{=}\frac{\mathrm{1}}{\pi }\int_{\mathrm{0} }^{\infty }\left|H\left ( \omega \right ) \right|^{\mathrm{2}}\psi _{x}\left ( \omega \right )d\omega }}$$
If the given linear system is an ideal low-pass filter with lower and upper cutoff frequencies f1 and f2 respectively. Then the magnitude of the system transfer function is
$$\mathrm{\mathit{\left|H\left ( \omega \right ) \right|\mathrm{=}\mathrm{1};\; \; \; \mathrm{for \; }f_{\mathrm{1}}< f< f_{\mathrm{2}}}}$$
Therefore, the energy of the output signal is,
$$\mathrm{\mathit{E_{y}\mathrm{=}\frac{\mathrm{1}}{\pi }\int_{f_{\mathrm{1}}}^{f_{\mathrm{2}}}\psi _{x}\left (\omega \right )d\omega \mathrm{=}\frac{\mathrm{1}}{\pi }\int_{f_{\mathrm{1}}}^{f_{\mathrm{2}}}\psi _{x}\left ( \mathrm{2} \pi f\right )d\left ( \mathrm{2}\pi f \right )}}$$
$$\mathrm{\Rightarrow \mathit{E_{y}\mathrm{=}\mathrm{2}\int_{f_{\mathrm{1}}}^{f_{\mathrm{2}}}\psi _{x}\left (f \right )df\; \; \;\cdot \cdot \cdot \left ( \mathrm{6} \right ) }}$$
Equation (6) gives the energy of the output signal of a linear system in terms of ESD of the input signal.
Properties of Energy Spectral Density
The properties of energy spectral density (ESD) are given as follows −
Property 1 – If $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is the input signal to a linear time-invariant system with impulse response h(t), then the energy spectral density (ESD) of the input and output signals are related as −
$$\mathrm{\mathit{\psi _{y}\left ( \omega \right )\mathrm{=}\left|H\left ( \omega \right ) \right|^{\mathrm{2}}\psi _{x}\left ( \omega \right )}}$$
Property 2 – The total area under the energy spectral density curve is equal to the total energy of the signal, i.e.,
$$\mathrm{\mathit{E\mathrm{=}\int_{-\infty }^{\infty }\psi \left ( f \right )df\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }\psi \left ( \omega \right )d\omega }}$$
Property 3 – The energy spectral density (ESD) function $\mathrm{\mathit{\psi \left ( \omega \right )}}$ and the autocorrelation function $\mathrm{\mathit{R\left ( \tau \right )}}$ of an energy signal form a Fourier transform pair, i.e.,
$$\mathrm{\mathit{R\left ( \tau \right )\overset{FT}{\leftrightarrow}\psi \left ( \omega \right )}}$$
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