Energy and power spectral density; spectrum analyzer

Energy spectral density (ESD)

For an energy signal $x(t)$ (finite energy, zero average power) with Fourier transform $X(f)$, the energy spectral density shows how the energy is spread over frequency:

$$\Psi_x(f) = |X(f)|^2 \quad (\text{J/Hz})$$

The total energy is the area under the ESD (Rayleigh's theorem):

$$E = \int_{-\infty}^{\infty} |x(t)|^2\,dt = \int_{-\infty}^{\infty} |X(f)|^2\,df$$

ESD is real, non-negative and even (for real $x(t)$), and it is the Fourier transform of the autocorrelation function $R_x(\tau)$.

Power spectral density (PSD)

For a power signal (finite non-zero average power, infinite energy) the Fourier transform may not exist, so we truncate the signal to $x_T(t)$ over $-T/2 < t < T/2$ and define

$$S_x(f) = \lim_{T\to\infty} \frac{|X_T(f)|^2}{T} \quad (\text{W/Hz})$$

The average power is $P = \int_{-\infty}^{\infty} S_x(f)\,df$. PSD is real, non-negative, even for real signals, and $S_x(f)$ and $R_x(\tau)$ form a Fourier transform pair (Wiener–Khinchine theorem).

Analog (swept-tuned) spectrum analyzer

A spectrum analyzer displays signal amplitude (or power) versus frequency on a screen. The common analog type is the swept-tuned superheterodyne analyzer.

 Input   +-------+   +-----+   +--------+   +-------+
 ------->| Atten |-->| LPF |-->| Mixer  |-->|  IF   |
         +-------+   +-----+   +--------+   | filter|
                                   ^        +-------+
                                   |            |
                              +---------+   +--------+
                              | Swept LO|   |Log amp |
                              |  (VCO)  |   |Env det |
                              +---------+   +--------+
                                   ^            |
                              +---------+   +--------+
                              |  Ramp   |   | Video  |
                              |generator|   | filter |
                              +---------+   +--------+
                                   |  X         | Y
                                   +-->[ DISPLAY ]<+

Operation:

1. Attenuator and low-pass filter (preselector): set a safe input level and remove image frequencies.
2. Ramp (sweep) generator: produces a sawtooth voltage. It tunes the local oscillator and also moves the display trace horizontally, so the X-axis becomes frequency.
3. Mixer and swept LO: the LO frequency is swept; a component of the input at $f_{in}$ is converted to the fixed IF when $|f_{LO} - f_{in}| = f_{IF}$. So, at each instant, only one input frequency reaches the IF stage.
4. IF filter: a narrow band-pass filter; its bandwidth is the resolution bandwidth (RBW), which decides how closely two spectral lines can be separated.
5. Log amplifier and envelope detector: convert the IF signal amplitude to a DC level, usually on a dB scale.
6. Video filter: smooths the detected output to reduce noise on the display.
7. Display: the vertical (Y) deflection is the detected amplitude and the horizontal (X) deflection is the ramp, so the screen shows amplitude versus frequency.

As the LO sweeps once over its range, each spectral component of the input appears as a peak at its frequency position, giving the amplitude spectrum (or power spectrum) of the signal.