Appendix B

Math Reference

The formulas behind the display, collected in one place. Fourier transform, window functions, probability of intercept, phase noise, decibels, Friis noise, EVM, and intermodulation.

Fourier Transform Cheat Sheet

Continuous Fourier Transform:

X(f) = \int_{-\infty}^{\infty} x(t) e^{-j 2\pi f t} dt

Discrete Fourier Transform:

X[k] = \sum_{n=0}^{N-1} x[n] e^{-j 2\pi k n / N}, \quad k = 0, 1, \ldots, N-1

Inverse DFT:

x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j 2\pi k n / N}

Bin frequency: $f_k = k \cdot f_s / N$. Resolution bandwidth: $\text{RBW} = f_s / N$.

Window Function Formulas

Hann (Hanning):

w[n] = 0.5 \left(1 - \cos\left(\frac{2\pi n}{N-1}\right)\right)

Hamming:

w[n] = 0.54 - 0.46 \cos\left(\frac{2\pi n}{N-1}\right)

Blackman:

w[n] = 0.42 - 0.5 \cos\left(\frac{2\pi n}{N-1}\right) + 0.08 \cos\left(\frac{4\pi n}{N-1}\right)

Blackman-Harris (4-term):

w[n] = 0.35875 - 0.48829 \cos\theta + 0.14128 \cos 2\theta - 0.01168 \cos 3\theta

where $\theta = 2\pi n / (N-1)$.

Equivalent Noise Bandwidth (ENBW):

\text{ENBW} = \frac{N \sum_n w[n]^2}{(\sum_n w[n])^2} \cdot \Delta f

Window	ENBW factor	Highest sidelobe (dB)	Main lobe -3 dB (bins)
Rectangular	1.00	-13	0.89
Hann	1.50	-32	1.44
Hamming	1.36	-43	1.30
Blackman	1.73	-58	1.68
Blackman-Harris	2.00	-92	1.90
Flat-top	3.77	-88	3.86

Probability of Intercept (Swept Analyzer) Derivation

Given sweep time $T_{\text{sweep}}$, span $\Delta f$, IF filter bandwidth $B_{IF}$ (roughly equal to RBW), and burst duration $t_b$, the dwell time at any single frequency bin is:

t_{\text{dwell}} = \frac{B_{IF} \cdot T_{\text{sweep}}}{\Delta f}

The probability of intercept per sweep (assuming uncorrelated burst timing) is:

P_{\text{intercept}} = \frac{t_{\text{dwell}} + t_b}{T_{\text{sweep}}}

For an RTSA with overlapping FFTs, $P_{\text{intercept}} \approx 1$ for any burst longer than the POI floor, which is set by FFT length, overlap ratio, and hardware peak detection.

Phase Noise Math

Phase noise as dBc/Hz at offset $f$ is written $L(f)$. Single-sideband phase noise is what is typically published. Total integrated phase noise over a frequency band is:

\sigma_\phi = \sqrt{2 \int_{f_1}^{f_2} 10^{L(f)/10} df}

Reciprocal mixing: a signal at offset $\Delta f$ from a strong carrier appears at amplitude reduced by $L(\Delta f) \cdot \text{ENBW}$ from phase noise alone.

Decibel Tables

Power ratios:

Linear	dB
0.001	-30
0.01	-20
0.1	-10
0.5	-3
1	0
2	+3
10	+10
100	+20
1000	+30
1,000,000	+60

dBm to milliwatts:

dBm	mW
-100	1e-13
-50	1e-8
-30	0.001
0	1
+10	10
+30	1000

Conversion formulas:

Power ratio: $\text{dB} = 10 \log_{10}(P_2 / P_1)$
Voltage/amplitude ratio: $\text{dB} = 20 \log_{10}(V_2 / V_1)$
dBm to watts: $W = 10^{(\text{dBm}-30)/10}$
dBm to mW: $\text{mW} = 10^{\text{dBm}/10}$

Friis Noise Formula

For a cascade of stages with noise factors $F_n$ and gains $G_n$ (linear):

F_{\text{total}} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots

Noise figure in dB: $\text{NF} = 10 \log_{10}(F)$.

EVM Math

\text{EVM}_{rms} = \sqrt{\frac{\frac{1}{N} \sum_n |\vec{e}[n]|^2}{P_{\text{ref}}}}

\text{EVM}_{dB} = 20 \log_{10}(\text{EVM}_{rms}) \qquad \text{MER}_{dB} = -\text{EVM}_{dB}

IIP3 and Intermodulation

For two equal-amplitude tones at input power $P$:

P_{IM3} = 3P - 2 \cdot P_{IIP3}

A 6 dB increase in input causes an 18 dB increase in IM3 products.