signal-processing-ai999x — offline, verified signal processing solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Engineering · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

signal-processing-ai999x is an offline, deterministic Python solver that answers signal processing questions with verified results from 74 certified capabilities, gives 334 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

signal-processing-ai999x is a compact, deterministic signal processing engine for Python. It reads a question in plain English (or a typed call), picks a certified signal processing capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified signal processing capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 334 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: sampling, Fourier series and transforms, the DFT and FFT, filters, convolution, z-transforms, modulation, communication links, RF transmission lines and radar.

Model overview

Property	Value
Package	signal-processing-ai999x 1.0.0 (import signal_processing_ai999x)
Type	Deterministic neuro-symbolic signal processing engine (rule-based reading + certified symbolic capabilities)
Capabilities	74 certified procedures
Knowledge pack	version 1.0.0
Golden answers	334 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install signal-processing-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/signal-processing-ai999x/resolve/main/dist/signal_processing_ai999x-1.0.0-py3-none-any.whl

import signal_processing_ai999x as signal_processing

signal_processing.ask("What is the Nyquist rate for a signal with a maximum frequency of 4 kHz?").show()
# -> SOLVED — fs_min = 8000, Ts_max = 0.000125  (nyquist_rate)
signal_processing.ask("Compute the DFT of x = [1, 2, 3, 4].").show()
# -> SOLVED — peak_magnitude = 10, peak_bin = 0, magnitude = [10.0, 2.8284271247461903, 2.0, 2.8284271247461903], phase = [0.0, 2.356194490192345, 3.141592653589793, -2.356194490192345]  (dft_spectrum)
signal_processing.ask("What is the bit error rate of BPSK at Eb/N0 = 8 dB?").show()
# -> SOLVED — ber = 0.000190908, ber_bfsk = 0.00600439  (bpsk_ber)
signal_processing.ask("A 50 ohm line is terminated in a load of 75 + j25 ohm. Find the reflection coefficient and the VSWR.").show()
# -> SOLVED — gamma_mag = 0.27735, gamma_angle_deg = 33.6901, gamma_re = 0.230769, gamma_im = 0.153846  (reflection_coefficient)
signal_processing.solve("butterworth_design", N=2, fc="1 kHz", fs="8 kHz", btype="lowpass").show()
# -> SOLVED — b = [0.0976310729378, 0.195262145876, 0.0976310729378], a = [1.0, -0.942809041582, 0.333333333333]  (butterworth_design)
signal_processing.ask("Who created you?").message
# -> signal-processing-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: signal-processing-ai999x "<question>", signal-processing-ai999x --capabilities. The outputs above were produced by running this exact code on signal-processing-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

74 certified procedures. A few of them:

• Procedures: AM modulation index from the envelope; Alias (apparent) frequency after sampling; Amplitude (voltage) ratio to decibels; Analytic signal, Hilbert transform and envelope; Bilinear transform of a first-order low-pass filter; Bit and symbol error rates of QPSK in AWGN; Bit error rate of BPSK (and coherent BFSK) in AWGN; Bit error rates of DPSK and noncoherent BFSK; Characteristic impedance of a coaxial line; Circular (cyclic) convolution; Coefficients of a window function; Cross-correlation and the lag of its peak; DC gain of a digital filter; DFT spectrum of a sequence (FFT); Decibels to amplitude (voltage) ratio; Decibels to power ratio; Digital Butterworth filter design (bilinear transform); Doppler shift of a radar echo; FIR filter output; FIR low-pass filter design by the window method; FM spectrum: Carson bandwidth and Bessel sideband amplitudes; Fourier transform of a decaying exponential; Fourier transform of a rectangular pulse; Free-space path loss …

signal_processing_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 74 formulas

Show every formula, by topic

DFT and FFT

DFT spectrum of a sequence (FFT)

$$X[k]=\sum _{n=0}^{N-1}x[n]e^{-j2\pi kn\mathrm{/}N},k=0,\dots ,N-1X[k]\; =\; \backslash sum\_\{n=0\}^\{N-1\}\; x[n]\backslash ,e^\{-j2\backslash pi\; kn/N\},\backslash \; k\; =\; 0,\backslash dots,N-1$$

Frequency of a DFT bin

$$f_k=\frac{k{f}_s}{N}f\_k\; =\; \backslash frac\{k\backslash ,f\_s\}\{N\}$$

Frequency resolution of an N-point DFT

$$\mathrm{\Delta }f=\frac{f_s}{N}\backslash Delta\; f\; =\; \backslash frac\{f\_s\}\{N\}$$

Inverse discrete Fourier transform

$$x[n]=\frac{1}{N}\sum _{k=0}^{N-1}X[k]e^{j2\pi kn\mathrm{/}N}x[n]\; =\; \backslash frac\{1\}\{N\}\backslash sum\_\{k=0\}^\{N-1\}\; X[k]\backslash ,e^\{j2\backslash pi\; kn/N\}$$

One bin of the discrete Fourier transform

$$X[k]=\sum _{n=0}^{N-1}x[n]e^{-j2\pi kn\mathrm{/}N}X[k]\; =\; \backslash sum\_\{n=0\}^\{N-1\}\; x[n]\backslash ,e^\{-j2\backslash pi\; kn/N\}$$

Fourier series and transforms

Analytic signal, Hilbert transform and envelope

$$x_a[n]=\mathrm{I}\mathrm{D}\mathrm{F}\mathrm{T}\{X[k]h[k]\},\text{envelope}=\mathrm{\mid }{x}_a[n]\mathrm{\mid }x\_a[n]\; =\; \backslash mathrm\{IDFT\}\backslash \{X[k]\backslash ,h[k]\backslash \},\backslash quad\; \backslash text\{envelope\}\; =\; \backslash vert\; x\_a[n]\backslash vert$$

Fourier transform of a decaying exponential

$$X(f)=\frac{A\tau }{1+j2\pi f\tau }X(f)\; =\; \backslash frac\{A\backslash tau\}\{1\; +\; j2\backslash pi\; f\backslash tau\}$$

Fourier transform of a rectangular pulse

$$X(f)=A\tau \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}(f\tau )=A\tau \frac{\sin (\pi f\tau )}{\pi f\tau }X(f)\; =\; A\backslash tau\backslash ,\backslash mathrm\{sinc\}(f\backslash tau)\; =\; A\backslash tau\backslash frac\{\backslash sin(\backslash pi\; f\backslash tau)\}\{\backslash pi\; f\backslash tau\}$$

Harmonic amplitude of a square, triangle or sawtooth wave

$$c_n^{\mathrm{s}\mathrm{q}}=\frac{4A}{n\pi },c_n^{\mathrm{t}\mathrm{r}\mathrm{i}}=\frac{8A}{{\pi }^2{n}^2}(n\text{ odd}),c_n^{\mathrm{s}\mathrm{a}\mathrm{w}}=\frac{2A}{n\pi }c\_n^\{\backslash mathrm\{sq\}\}\; =\; \backslash frac\{4A\}\{n\backslash pi\},\backslash \; c\_n^\{\backslash mathrm\{tri\}\}\; =\; \backslash frac\{8A\}\{\backslash pi^2\; n^2\}\backslash \; (n\backslash \; \backslash text\{odd\}),\backslash quad\; c\_n^\{\backslash mathrm\{saw\}\}\; =\; \backslash frac\{2A\}\{n\backslash pi\}$$

Total harmonic distortion

$$\mathrm{T}\mathrm{H}\mathrm{D}=\frac{\sqrt{\sum _{n\ge 2}{V}_n^2}}{V_1}\backslash mathrm\{THD\}\; =\; \backslash frac\{\backslash sqrt\{\backslash sum\_\{n\backslash ge\; 2\}\; V\_n^2\}\}\{V\_1\}$$

Analog filters

Magnitude of an analog Butterworth low-pass response

$$\mathrm{\mid }H(f)\mathrm{\mid }=\frac{1}{\sqrt{1+(f\mathrm{/}{f}_c{)}^{2N}}}\backslash vert\; H(f)\backslash vert\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\; +\; (f/f\_c)^\{2N\}\}\}$$

Minimum order of a analog Butterworth filter

$$N\ge \frac{{\log }_{10}\frac{{10}^{A_s\mathrm{/}10}-1}{{10}^{R_p\mathrm{/}10}-1}}{2{\log }_{10}({\omega }_s\mathrm{/}{\omega }_p)}N\; \backslash ge\; \backslash frac\{\backslash log\_\{10\}\backslash frac\{10^\{A\_s/10\}-1\}\{10^\{R\_p/10\}-1\}\}\{2\backslash log\_\{10\}(\backslash omega\_s/\backslash omega\_p)\}$$

Minimum order of a analog Chebyshev type I filter

$$N\ge \frac{{\cosh }^{-1}\sqrt{\frac{{10}^{A_s\mathrm{/}10}-1}{{10}^{R_p\mathrm{/}10}-1}}}{{\cosh }^{-1}({\omega }_s\mathrm{/}{\omega }_p)}N\; \backslash ge\; \backslash frac\{\backslash cosh^\{-1\}\backslash sqrt\{\backslash frac\{10^\{A\_s/10\}-1\}\{10^\{R\_p/10\}-1\}\}\}\{\backslash cosh^\{-1\}(\backslash omega\_s/\backslash omega\_p)\}$$

Antennas

Gain of a parabolic dish (aperture) antenna

$$G=\eta {\left(\frac{\pi D}{\lambda }\right)}^{2}G\; =\; \backslash eta\backslash left(\backslash frac\{\backslash pi\; D\}\{\backslash lambda\}\backslash right)^2$$

Communication links

Minimum SNR for a data rate (Shannon limit)

$${\mathrm{S}\mathrm{N}\mathrm{R}}_{\min }=2^{C\mathrm{/}B}-1,{\left(\frac{E_b}{N_0}\right)}_{\min }=\frac{2^{C\mathrm{/}B}-1}{C\mathrm{/}B}\backslash mathrm\{SNR\}\_\{\backslash min\}\; =\; 2^\{C/B\}\; -\; 1,\backslash quad\; \backslash left(\backslash tfrac\{E\_b\}\{N\_0\}\backslash right)\_\{\backslash min\}\; =\; \backslash frac\{2^\{C/B\}-1\}\{C/B\}$$

Shannon capacity of an AWGN channel

$$C=B{\log }_2(1+\mathrm{S}\mathrm{N}\mathrm{R})C\; =\; B\backslash log\_2(1\; +\; \backslash mathrm\{SNR\})$$

Convolution and correlation

Circular (cyclic) convolution

$$y[n]=\sum _{k=0}^{N-1}x[k]h[(n-k)N]y[n]\; =\; \backslash sum\_\{k=0\}^\{N-1\}\; x[k]\backslash ,h[(n-k)\backslash bmod\; N]$$

Cross-correlation and the lag of its peak

$$r_{xy}[\mathrm{\ell }]=\sum _{n}x[n+\mathrm{\ell }]y[n]r\_\{xy\}[\backslash ell]\; =\; \backslash sum\_n\; x[n+\backslash ell]\backslash ,y[n]$$

Linear convolution of two sequences

$$y[n]=\sum _{k}x[k]h[n-k]y[n]\; =\; \backslash sum\_k\; x[k]\backslash ,h[n-k]$$

Decibels and levels

Amplitude (voltage) ratio to decibels

$$L=20{\log }_{10}\frac{V_2}{V_1}\mathrm{d}\mathrm{B}L\; =\; 20\backslash log\_\{10\}\backslash frac\{V\_2\}\{V\_1\}\backslash \; \backslash mathrm\{dB\}$$

Decibels to amplitude (voltage) ratio

$$\frac{V_2}{V_1}={10}^{L\mathrm{/}20}\backslash frac\{V\_2\}\{V\_1\}\; =\; 10^\{L/20\}$$

Decibels to power ratio

$$\frac{P_2}{P_1}={10}^{L\mathrm{/}10}\backslash frac\{P\_2\}\{P\_1\}\; =\; 10^\{L/10\}$$

Power in watts to dBm and dBW

$$P_{\mathrm{d}\mathrm{B}\mathrm{m}}=10{\log }_{10}\frac{P}{1\mathrm{m}\mathrm{W}}P\_\{\backslash mathrm\{dBm\}\}\; =\; 10\backslash log\_\{10\}\backslash frac\{P\}\{1\backslash ,\backslash mathrm\{mW\}\}$$

Power level in dBm to watts

$$P=1\mathrm{m}\mathrm{W}\cdot {10}^{P_{\mathrm{d}\mathrm{B}\mathrm{m}}\mathrm{/}10}P\; =\; 1\backslash ,\backslash mathrm\{mW\}\backslash cdot\; 10^\{P\_\{\backslash mathrm\{dBm\}\}/10\}$$

Power ratio to decibels

$$L=10{\log }_{10}\frac{P_2}{P_1}\mathrm{d}\mathrm{B}L\; =\; 10\backslash log\_\{10\}\backslash frac\{P\_2\}\{P\_1\}\backslash \; \backslash mathrm\{dB\}$$

Signal-to-noise ratio from signal and noise powers

$${\mathrm{S}\mathrm{N}\mathrm{R}}_{\mathrm{d}\mathrm{B}}=10{\log }_{10}\frac{P_s}{P_n}\backslash mathrm\{SNR\}\_\{\backslash mathrm\{dB\}\}\; =\; 10\backslash log\_\{10\}\backslash frac\{P\_s\}\{P\_n\}$$

Total power of several signals given in dBm

$$P_{\mathrm{t}\mathrm{o}\mathrm{t}}=10{\log }_{10}\sum _i{10}^{P_i\mathrm{/}10}\mathrm{d}\mathrm{B}\mathrm{m}P\_\{\backslash mathrm\{tot\}\}\; =\; 10\backslash log\_\{10\}\backslash sum\_i\; 10^\{P\_i/10\}\backslash \; \backslash mathrm\{dBm\}$$

Digital communication

Bit and symbol error rates of QPSK in AWGN

$$P_b=Q\text{\hspace{0.17em}⁣}\left(\sqrt{2E_b\mathrm{/}{N}_0}\right),P_s=2Q-Q^{2}P\_b\; =\; Q\backslash !\backslash left(\backslash sqrt\{2E\_b/N\_0\}\backslash right),\backslash quad\; P\_s\; =\; 2Q\; -\; Q^2$$

Bit error rate of BPSK (and coherent BFSK) in AWGN

$$P_b=Q\text{\hspace{0.17em}⁣}\left(\sqrt{2E_b\mathrm{/}{N}_0}\right)=\frac{1}{2}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\sqrt{E_b\mathrm{/}{N}_0}P\_b\; =\; Q\backslash !\backslash left(\backslash sqrt\{2E\_b/N\_0\}\backslash right)\; =\; \backslash tfrac12\backslash ,\backslash mathrm\{erfc\}\backslash sqrt\{E\_b/N\_0\}$$

Bit error rates of DPSK and noncoherent BFSK

$$P_b^{\mathrm{D}\mathrm{P}\mathrm{S}\mathrm{K}}=\frac{1}{2}{e}^{-E_b\mathrm{/}{N}_0},P_b^{\mathrm{N}\mathrm{C}\mathrm{F}\mathrm{S}\mathrm{K}}=\frac{1}{2}{e}^{-E_b\mathrm{/}2N_0}P\_b^\{\backslash mathrm\{DPSK\}\}\; =\; \backslash tfrac12\; e^\{-E\_b/N\_0\},\backslash quad\; P\_b^\{\backslash mathrm\{NCFSK\}\}\; =\; \backslash tfrac12\; e^\{-E\_b/2N\_0\}$$

Symbol error rate of M-PSK in AWGN

$$P_s=\frac{1}{\pi }{\int }_0^{(M-1)\pi \mathrm{/}M}\exp \text{\hspace{0.17em}⁣}(-\frac{E_s}{N_0}\frac{{\sin }^2(\pi \mathrm{/}M)}{{\sin }^2\theta })d\theta P\_s\; =\; \backslash frac\{1\}\{\backslash pi\}\backslash int\_0^\{(M-1)\backslash pi/M\}\; \backslash exp\backslash !\backslash left(-\backslash frac\{E\_s\}\{N\_0\}\backslash frac\{\backslash sin^2(\backslash pi/M)\}\{\backslash sin^2\backslash theta\}\backslash right)d\backslash theta$$

Symbol error rate of square M-QAM in AWGN

$$P_s=1-{[1-2(1-\frac{1}{\sqrt{M}})Q\text{\hspace{0.17em}⁣}\left(\sqrt{\frac{3E_s}{(M-1)N_0}}\right)]}^{2}P\_s\; =\; 1\; -\; \backslash left[1\; -\; 2\backslash left(1\; -\; \backslash tfrac\{1\}\{\backslash sqrt\; M\}\backslash right)Q\backslash !\backslash left(\backslash sqrt\{\backslash tfrac\{3E\_s\}\{(M-1)N\_0\}\}\backslash right)\backslash right]^2$$

Digital filters

Bilinear transform of a first-order low-pass filter

$$H(z)=\frac{{\omega }_c(1+z^{-1})}{({\omega }_c+2f_s)+({\omega }_c-2f_s)z^{-1}}H(z)\; =\; \backslash frac\{\backslash omega\_c\; (1\; +\; z^\{-1\})\}\{(\backslash omega\_c\; +\; 2f\_s)\; +\; (\backslash omega\_c\; -\; 2f\_s)\; z^\{-1\}\}$$

DC gain of a digital filter

$$H(1)=\frac{\sum _k{b}_k}{\sum _k{a}_k}H(1)\; =\; \backslash frac\{\backslash sum\_k\; b\_k\}\{\backslash sum\_k\; a\_k\}$$

FIR filter output

$$y[n]=\sum _{k=0}^M{b}_{k}x[n-k]y[n]\; =\; \backslash sum\_\{k=0\}^\{M\}\; b\_k\backslash ,x[n-k]$$

Frequency response of a digital filter at one frequency

$$H(e^{j\omega })=\frac{\sum _k{b}_k{e}^{-j\omega k}}{\sum _k{a}_k{e}^{-j\omega k}},\omega =2\pi f\mathrm{/}{f}_{s}H(e^\{j\backslash omega\})\; =\; \backslash frac\{\backslash sum\_k\; b\_k\; e^\{-j\backslash omega\; k\}\}\{\backslash sum\_k\; a\_k\; e^\{-j\backslash omega\; k\}\},\backslash \; \backslash omega\; =\; 2\backslash pi\; f/f\_s$$

Group delay of a digital filter

$${\tau }_g(\omega )=-\frac{d\mathrm{\angle }H(e^{j\omega })}{d\omega }=\mathrm{R}\mathrm{e}\frac{\sum k{b}_k{e}^{-j\omega k}}{\sum b_k{e}^{-j\omega k}}-\mathrm{R}\mathrm{e}\frac{\sum k{a}_k{e}^{-j\omega k}}{\sum a_k{e}^{-j\omega k}}\backslash tau\_g(\backslash omega)\; =\; -\backslash frac\{d\backslash angle\; H(e^\{j\backslash omega\})\}\{d\backslash omega\}\; =\; \backslash mathrm\{Re\}\backslash frac\{\backslash sum\; k\; b\_k\; e^\{-j\backslash omega\; k\}\}\{\backslash sum\; b\_k\; e^\{-j\backslash omega\; k\}\}\; -\; \backslash mathrm\{Re\}\backslash frac\{\backslash sum\; k\; a\_k\; e^\{-j\backslash omega\; k\}\}\{\backslash sum\; a\_k\; e^\{-j\backslash omega\; k\}\}$$

IIR filter output (difference equation)

$$a_{0}y[n]=\sum _k{b}_{k}x[n-k]-\sum _{k\ge 1}{a}_{k}y[n-k]a\_0\; y[n]\; =\; \backslash sum\_k\; b\_k\; x[n-k]\; -\; \backslash sum\_\{k\backslash ge\; 1\}\; a\_k\; y[n-k]$$

Impulse response of a digital filter

$$h[n]={\mathcal{Z}}^{-1}\{B(z)\mathrm{/}A(z)\}h[n]\; =\; \backslash mathcal\{Z\}^\{-1\}\backslash \{B(z)/A(z)\backslash \}$$

Minimum order of a digital (bilinear) Butterworth filter

$$N\ge \frac{{\log }_{10}\frac{{10}^{A_s\mathrm{/}10}-1}{{10}^{R_p\mathrm{/}10}-1}}{2{\log }_{10}({\omega }_s\mathrm{/}{\omega }_p)}N\; \backslash ge\; \backslash frac\{\backslash log\_\{10\}\backslash frac\{10^\{A\_s/10\}-1\}\{10^\{R\_p/10\}-1\}\}\{2\backslash log\_\{10\}(\backslash omega\_s/\backslash omega\_p)\}$$

Moving-average filter output

$$y[n]=\frac{1}{M}\sum _{i=0}^{M-1}x[n-i]y[n]\; =\; \backslash frac\{1\}\{M\}\backslash sum\_\{i=0\}^\{M-1\}\; x[n-i]$$

Step response of a digital filter

$$s[n]=\sum _{m=0}^{n}h[m]s[n]\; =\; \backslash sum\_\{m=0\}^\{n\}\; h[m]$$

Filter design

Digital Butterworth filter design (bilinear transform)

$$s=2f_s\frac{1-z^{-1}}{1+z^{-1}},{\omega }_c=2f_s\tan \frac{\pi f_c}{f_s}s\; =\; 2f\_s\backslash frac\{1\; -\; z^\{-1\}\}\{1\; +\; z^\{-1\}\},\backslash quad\; \backslash omega\_c\; =\; 2f\_s\backslash tan\backslash frac\{\backslash pi\; f\_c\}\{f\_s\}$$

FIR low-pass filter design by the window method

$$h[n]\propto \frac{2f_c}{f_s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\text{\hspace{0.17em}⁣}\left(\frac{2f_c}{f_s}(n-\frac{N-1}{2})\right)w[n]h[n]\; \backslash propto\; \backslash frac\{2f\_c\}\{f\_s\}\backslash ,\backslash mathrm\{sinc\}\backslash !\backslash left(\backslash frac\{2f\_c\}\{f\_s\}\backslash left(n\; -\; \backslash tfrac\{N-1\}\{2\}\backslash right)\backslash right)\; w[n]$$

Second-order IIR notch filter design

$$H(z)=g\frac{1-2\cos {\omega }_0{z}^{-1}+z^{-2}}{1-2g\cos {\omega }_0{z}^{-1}+(2g-1)z^{-2}},g=\frac{1}{1+\tan ({\omega }_0\mathrm{/}2Q)}H(z)\; =\; g\backslash ,\backslash frac\{1\; -\; 2\backslash cos\backslash omega\_0\; z^\{-1\}\; +\; z^\{-2\}\}\{1\; -\; 2g\backslash cos\backslash omega\_0\; z^\{-1\}\; +\; (2g\; -\; 1)z^\{-2\}\},\backslash \; g\; =\; \backslash frac\{1\}\{1\; +\; \backslash tan(\backslash omega\_0/2Q)\}$$

Modulation

AM modulation index from the envelope

$$m=\frac{V_{\max }-V_{\min }}{V_{\max }+V_{\min }}m\; =\; \backslash frac\{V\_\{\backslash max\}\; -\; V\_\{\backslash min\}\}\{V\_\{\backslash max\}\; +\; V\_\{\backslash min\}\}$$

FM spectrum: Carson bandwidth and Bessel sideband amplitudes

$$B_C=2(\mathrm{\Delta }f+f_m),s(t)=A\sum _n{J}_n(\beta )\cos [2\pi (f_c+n{f}_m)t]B\_C\; =\; 2(\backslash Delta\; f\; +\; f\_m),\backslash quad\; s(t)\; =\; A\backslash sum\_n\; J\_n(\backslash beta)\backslash cos[2\backslash pi(f\_c\; +\; n\; f\_m)t]$$

Power in an AM (DSB full-carrier) signal

$$P_t=P_c(1+\frac{m^2}{2})P\_t\; =\; P\_c\backslash left(1\; +\; \backslash frac\{m^2\}\{2\}\backslash right)$$

Noise and link budgets

Free-space path loss

$$L_{\mathrm{f}\mathrm{s}}={\left(\frac{4\pi df}{c}\right)}^{2}L\_\{\backslash mathrm\{fs\}\}\; =\; \backslash left(\backslash frac\{4\backslash pi\; d\; f\}\{c\}\backslash right)^2$$

Noise figure of cascaded stages (Friis formula)

$$F=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1{G}_2}+\cdots F\; =\; F\_1\; +\; \backslash frac\{F\_2\; -\; 1\}\{G\_1\}\; +\; \backslash frac\{F\_3\; -\; 1\}\{G\_1\; G\_2\}\; +\; \backslash cdots$$

Received power of a radio link (Friis transmission equation)

$$P_r=P_t{G}_t{G}_r{\left(\frac{\lambda }{4\pi d}\right)}^{2}P\_r\; =\; P\_t\; G\_t\; G\_r\; \backslash left(\backslash frac\{\backslash lambda\}\{4\backslash pi\; d\}\backslash right)^2$$

Radar

Doppler shift of a radar echo

$$f_d=\frac{2vf}{c-v}\approx \frac{2v}{\lambda }f\_d\; =\; \backslash frac\{2vf\}\{c\; -\; v\}\; \backslash approx\; \backslash frac\{2v\}\{\backslash lambda\}$$

Maximum detection range of a radar

$$R_{\max }={\left(\frac{P_t{G}^2{\lambda }^2\sigma }{(4\pi {)}^3{S}_{\min }}\right)}^{1\mathrm{/}4}R\_\{\backslash max\}\; =\; \backslash left(\backslash frac\{P\_t\; G^2\; \backslash lambda^2\; \backslash sigma\}\{(4\backslash pi)^3\; S\_\{\backslash min\}\}\backslash right)^\{1/4\}$$

Received echo power (radar range equation)

$$P_r=\frac{P_t{G}^2{\lambda }^2\sigma }{(4\pi {)}^3{R}^4}P\_r\; =\; \backslash frac\{P\_t\; G^2\; \backslash lambda^2\; \backslash sigma\}\{(4\backslash pi)^3\; R^4\}$$

Sampling and signal basics

Alias (apparent) frequency after sampling

$$f_a=\mid f-f_s\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}(f\mathrm{/}{f}_s)\mid f\_a\; =\; \backslash left\backslash vert\; f\; -\; f\_s\backslash ,\backslash mathrm\{round\}(f/f\_s)\; \backslash right\backslash vert$$

Frequency and angular frequency from period

$$f=\frac{1}{T},\omega =\frac{2\pi }{T}f\; =\; \backslash frac\{1\}\{T\},\backslash quad\; \backslash omega\; =\; \backslash frac\{2\backslash pi\}\{T\}$$

Mean, RMS, energy and average power of a sampled signal

$$E=\sum _n\mathrm{\mid }x[n]{\mathrm{\mid }}^2,P=\frac{E}{N},x_{\mathrm{r}\mathrm{m}\mathrm{s}}=\sqrt{P}E\; =\; \backslash sum\_n\; \backslash vert\; x[n]\backslash vert\; ^2,\backslash quad\; P\; =\; \backslash frac\{E\}\{N\},\backslash quad\; x\_\{\backslash mathrm\{rms\}\}\; =\; \backslash sqrt\{P\}$$

Nyquist rate

$$f_{s,\min }=2f_{\max }f\_\{s,\backslash min\}\; =\; 2\; f\_\{\backslash max\}$$

Period and angular frequency from frequency

$$T=\frac{1}{f},\omega =2\pi fT\; =\; \backslash frac\{1\}\{f\},\backslash quad\; \backslash omega\; =\; 2\backslash pi\; f$$

RMS, peak-to-peak and rectified average of a sine wave

$$V_{\mathrm{r}\mathrm{m}\mathrm{s}}=\frac{V_p}{\sqrt{2}}V\_\{\backslash mathrm\{rms\}\}\; =\; \backslash frac\{V\_p\}\{\backslash sqrt\{2\}\}$$

Signal-to-quantization-noise ratio of an ideal ADC

$$\mathrm{S}\mathrm{Q}\mathrm{N}\mathrm{R}=10{\log }_{10}\text{\hspace{0.17em}⁣}\left(\frac{3}{2}{2}^{2N}\right)\approx 6.02N+1.76\mathrm{d}\mathrm{B}\backslash mathrm\{SQNR\}\; =\; 10\backslash log\_\{10\}\backslash !\backslash left(\backslash tfrac\{3\}\{2\}\backslash ,2^\{2N\}\backslash right)\; \backslash approx\; 6.02N\; +\; 1.76\backslash \; \backslash mathrm\{dB\}$$

Transmission lines

Characteristic impedance of a coaxial line

$$Z_0=\frac{{\eta }_0}{2\pi \sqrt{{\epsilon }_r}}\ln \frac{D}{d}Z\_0\; =\; \backslash frac\{\backslash eta\_0\}\{2\backslash pi\backslash sqrt\{\backslash varepsilon\_r\}\}\backslash ln\backslash frac\{D\}\{d\}$$

Input impedance of a terminated lossless line

$$Z_{in}=Z_0\frac{Z_L+j{Z}_0\tan \beta l}{Z_0+j{Z}_L\tan \beta l}Z\_\{in\}\; =\; Z\_0\backslash frac\{Z\_L\; +\; jZ\_0\backslash tan\backslash beta\; l\}\{Z\_0\; +\; jZ\_L\backslash tan\backslash beta\; l\}$$

Quarter-wave impedance transformer

$$Z_t=\sqrt{R_L{Z}_0},\mathrm{\ell }=\frac{v_{f}c}{4f}Z\_t\; =\; \backslash sqrt\{R\_L\; Z\_0\},\backslash quad\; \backslash ell\; =\; \backslash frac\{v\_f\backslash ,c\}\{4f\}$$

Reflection coefficient and return loss from the VSWR

$$\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }=\frac{\mathrm{V}\mathrm{S}\mathrm{W}\mathrm{R}-1}{\mathrm{V}\mathrm{S}\mathrm{W}\mathrm{R}+1}\backslash vert\; \backslash Gamma\backslash vert\; =\; \backslash frac\{\backslash mathrm\{VSWR\}\; -\; 1\}\{\backslash mathrm\{VSWR\}\; +\; 1\}$$

Reflection coefficient, VSWR and return loss of a load

$$\mathrm{\Gamma }=\frac{Z_L-Z_0}{Z_L+Z_0},\mathrm{V}\mathrm{S}\mathrm{W}\mathrm{R}=\frac{1+\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }}{1-\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }}\backslash Gamma\; =\; \backslash frac\{Z\_L\; -\; Z\_0\}\{Z\_L\; +\; Z\_0\},\backslash quad\; \backslash mathrm\{VSWR\}\; =\; \backslash frac\{1\; +\; \backslash vert\; \backslash Gamma\backslash vert\; \}\{1\; -\; \backslash vert\; \backslash Gamma\backslash vert\; \}$$

VSWR and return loss from the reflection coefficient

$$\mathrm{V}\mathrm{S}\mathrm{W}\mathrm{R}=\frac{1+\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }}{1-\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }}\backslash mathrm\{VSWR\}\; =\; \backslash frac\{1\; +\; \backslash vert\; \backslash Gamma\backslash vert\; \}\{1\; -\; \backslash vert\; \backslash Gamma\backslash vert\; \}$$

Wavelets

Haar discrete wavelet transform

$$c_A[k]=\frac{x[2k]+x[2k+1]}{\sqrt{2}},c_D[k]=\frac{x[2k]-x[2k+1]}{\sqrt{2}}c\_A[k]\; =\; \backslash frac\{x[2k]\; +\; x[2k+1]\}\{\backslash sqrt\; 2\},\backslash quad\; c\_D[k]\; =\; \backslash frac\{x[2k]\; -\; x[2k+1]\}\{\backslash sqrt\; 2\}$$

Inverse Haar wavelet transform (one level)

$$x[2k]=\frac{c_A[k]+c_D[k]}{\sqrt{2}},x[2k+1]=\frac{c_A[k]-c_D[k]}{\sqrt{2}}x[2k]\; =\; \backslash frac\{c\_A[k]\; +\; c\_D[k]\}\{\backslash sqrt\; 2\},\backslash quad\; x[2k+1]\; =\; \backslash frac\{c\_A[k]\; -\; c\_D[k]\}\{\backslash sqrt\; 2\}$$

Windows and spectral analysis

Coefficients of a window function

$$w[n]=\sum _k(-1{)}^k{a}_k\cos \frac{2\pi kn}{M}w[n]\; =\; \backslash sum\_k\; (-1)^k\; a\_k\; \backslash cos\backslash frac\{2\backslash pi\; k\; n\}\{M\}$$

Main-lobe width of a spectral window

$$\mathrm{\Delta }{f}_{\mathrm{M}\mathrm{L}}=k\frac{f_s}{N},k=2(\text{rect}),4(\text{Hann, Hamming}),6(\text{Blackman})\backslash Delta\; f\_\{\backslash mathrm\{ML\}\}\; =\; k\backslash ,\backslash frac\{f\_s\}\{N\},\backslash \; k\; =\; 2\backslash \; (\backslash text\{rect\}),\backslash \; 4\backslash \; (\backslash text\{Hann,\; Hamming\}),\backslash \; 6\backslash \; (\backslash text\{Blackman\})$$

Periodogram (power spectral density estimate)

Z-transform and stability

Poles and stability of a second-order all-pole system

Poles, zeros and stability of a digital filter

$$H(z)=\frac{\sum _k{b}_k{z}^{-k}}{\sum _k{a}_k{z}^{-k}}=\frac{b_0\prod (1-z_i{z}^{-1})}{a_0\prod (1-p_i{z}^{-1})}H(z)\; =\; \backslash frac\{\backslash sum\_k\; b\_k\; z^\{-k\}\}\{\backslash sum\_k\; a\_k\; z^\{-k\}\}\; =\; \backslash frac\{b\_0\backslash prod(1\; -\; z\_i\; z^\{-1\})\}\{a\_0\backslash prod(1\; -\; p\_i\; z^\{-1\})\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified signal processing capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (sampling, Fourier series and transforms, the DFT and FFT, filters, convolution, z-transforms, modulation, communication links, RF transmission lines and radar); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{signal_processing_ai999x_1_0_0,
  title  = {signal-processing-ai999x: an offline, verified signal processing solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/signal-processing-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/signal-processing-ai999x/discussions or visit https://ai999x.com.

Support the project

signal-processing-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com