Generating Signal Processing Equations

Prompt

Create a list of the 25 most commonly used equations in a college level signal processing course.

For each equation, return the following:

The name of the equation in markdown level 4.
Within the body, return the equation in LaTeX format
Surround the equation with dollar signs
For each variable in the equation, add a "where:" that defines each variable
Put blank lines around the list of variables
Wrap the variables names in dollar signs

Examples:

Slope and Intercept

\(y = m * x + b\)

where:

\(y\) is the vertical axis
\(x\) is the horizontal axis
\(m\) is the slope
\(b\) is the intercept

Area of a Circle

\(A = 2\pi r^2\) where:

\(A\) is the area of the circle
\(r\) is the radius

After the first 25 equations were generated I used the following prompt:

Prompt

Excellent! Please continue with the next 25 most commonly used equations in a signal processing course.

Responses

Note that the 100 top equations are group in four ranges:

1-25
26-50
50-75
75-100

Equations 1-25

Discrete Fourier Transform (DFT)

\(X[k] = \sum_{n=0}^{N-1} x[n]e^{-j2\pi kn/N}\)

where:

\(X[k]\) is the frequency domain representation
\(x[n]\) is the time domain signal
\(N\) is the number of samples
\(k\) is the frequency index
\(n\) is the time index

Inverse Discrete Fourier Transform (IDFT)

\(x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k]e^{j2\pi kn/N}\)

where:

\(x[n]\) is the reconstructed time domain signal
\(X[k]\) is the frequency domain representation
\(N\) is the number of samples
\(k\) is the frequency index
\(n\) is the time index

Sampling Theorem (Nyquist Rate)

\(f_s \geq 2f_{max}\)

where:

\(f_s\) is the sampling frequency
\(f_{max}\) is the highest frequency component in the signal

Z-Transform

\(X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}\)

where:

\(X(z)\) is the z-domain representation
\(x[n]\) is the time domain signal
\(z\) is the complex variable
\(n\) is the time index

Inverse Z-Transform

\(x[n] = \frac{1}{2\pi j}\oint_C X(z)z^{n-1}dz\)

where:

\(x[n]\) is the time domain signal
\(X(z)\) is the z-domain representation
\(C\) is the contour of integration
\(z\) is the complex variable

Convolution Sum

\(y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k]\)

where:

\(y[n]\) is the output signal
\(x[n]\) is the input signal
\(h[n]\) is the impulse response
\(k\) is the summation index

Correlation

\(R_{xy}[n] = \sum_{k=-\infty}^{\infty} x[k]y[k+n]\)

where:

\(R_{xy}[n]\) is the correlation sequence
\(x[k]\) is the first signal
\(y[k]\) is the second signal
\(n\) is the lag variable

Power Spectral Density

\(P_{xx}(e^{j\omega}) = \sum_{n=-\infty}^{\infty} R_{xx}[n]e^{-j\omega n}\)

where:

\(P_{xx}(e^{j\omega})\) is the power spectral density
\(R_{xx}[n]\) is the autocorrelation sequence
\(\omega\) is the angular frequency
\(n\) is the lag variable

First Difference Equation

\(y[n] = y[n-1] + x[n]\)

where:

\(y[n]\) is the output signal
\(x[n]\) is the input signal
\(n\) is the time index

Moving Average Filter

\(y[n] = \frac{1}{M}\sum_{k=0}^{M-1} x[n-k]\)

where:

\(y[n]\) is the filtered output
\(x[n]\) is the input signal
\(M\) is the number of points in the average
\(k\) is the summation index

IIR Filter General Form

\(y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k]\)

where:

\(y[n]\) is the filter output
\(x[n]\) is the filter input
\(b_k\) are the feedforward coefficients
\(a_k\) are the feedback coefficients
\(M\) is the feedforward filter order
\(N\) is the feedback filter order

Frequency Response

\(H(e^{j\omega}) = \sum_{n=-\infty}^{\infty} h[n]e^{-j\omega n}\)

where:

\(H(e^{j\omega})\) is the frequency response
\(h[n]\) is the impulse response
\(\omega\) is the angular frequency
\(n\) is the time index

Phase Response

\(\phi(\omega) = \tan^{-1}\left(\frac{\Im\{H(e^{j\omega})\}}{\Re\{H(e^{j\omega})\}}\right)\)

where:

\(\phi(\omega)\) is the phase response
\(H(e^{j\omega})\) is the frequency response
\(\omega\) is the angular frequency
\(\Im\) represents the imaginary part
\(\Re\) represents the real part

Magnitude Response

\(|H(e^{j\omega})| = \sqrt{\Re\{H(e^{j\omega})\}^2 + \Im\{H(e^{j\omega})\}^2}\)

where:

\(|H(e^{j\omega})|\) is the magnitude response
\(H(e^{j\omega})\) is the frequency response
\(\omega\) is the angular frequency
\(\Im\) represents the imaginary part
\(\Re\) represents the real part

Group Delay

\(\tau_g(\omega) = -\frac{d\phi(\omega)}{d\omega}\)

where:

\(\tau_g(\omega)\) is the group delay
\(\phi(\omega)\) is the phase response
\(\omega\) is the angular frequency

Parseval's Theorem

\(\sum_{n=-\infty}^{\infty} |x[n]|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi} |X(e^{j\omega})|^2 d\omega\)

where:

\(x[n]\) is the time domain signal
\(X(e^{j\omega})\) is the frequency domain representation
\(\omega\) is the angular frequency

Discrete-Time Fourier Transform (DTFT)

\(X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n}\)

where:

\(X(e^{j\omega})\) is the frequency domain representation
\(x[n]\) is the time domain signal
\(\omega\) is the angular frequency
\(n\) is the time index

Window Function (Hamming)

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

where:

\(w[n]\) is the window function value
\(n\) is the sample index
\(N\) is the window length

Discrete Cosine Transform (DCT)

\(X[k] = \sum_{n=0}^{N-1} x[n]\cos[\frac{\pi k(2n+1)}{2N}]\)

where:

\(X[k]\) is the DCT coefficient
\(x[n]\) is the input signal
\(N\) is the number of samples
\(k\) is the frequency index
\(n\) is the time index

Energy of a Signal

\(E = \sum_{n=-\infty}^{\infty} |x[n]|^2\)

where:

\(E\) is the total energy
\(x[n]\) is the signal
\(n\) is the time index

Power of a Signal

\(P = \lim_{N\to\infty} \frac{1}{2N+1}\sum_{n=-N}^{N} |x[n]|^2\)

where:

\(P\) is the average power
\(x[n]\) is the signal
\(N\) is the number of samples

Decimation Equation

\(y[n] = x[Mn]\)

where:

\(y[n]\) is the decimated signal
\(x[n]\) is the input signal
\(M\) is the decimation factor
\(n\) is the time index

Interpolation Equation

\(y[n] = \begin{cases} x[n/L] & \text{if } n/L \text{ is integer} \\ 0 & \text{otherwise} \end{cases}\)

where:

\(y[n]\) is the interpolated signal
\(x[n]\) is the input signal
\(L\) is the interpolation factor
\(n\) is the time index

Signal-to-Noise Ratio (SNR)

\(SNR = 10\log_{10}\left(\frac{P_{signal}}{P_{noise}}\right)\)

where:

\(SNR\) is the signal-to-noise ratio in dB
\(P_{signal}\) is the power of the signal
\(P_{noise}\) is the power of the noise

Circular Convolution

\(y[n] = \sum_{k=0}^{N-1} x[k]h[(n-k)_N]\)

where:

\(y[n]\) is the output signal
\(x[n]\) is the input signal
\(h[n]\) is the impulse response
\(N\) is the period of the circular convolution
\((n-k)_N\) denotes modulo-N operation

I'll continue with the next 25 common signal processing equations following the same format.

Equations 25-50

Fast Fourier Transform (Radix-2 Decimation in Time)

\(X[k] = \sum_{n=0}^{N/2-1} x[2n]W_N^{2nk} + W_N^k\sum_{n=0}^{N/2-1} x[2n+1]W_N^{2nk}\)

where:

\(X[k]\) is the frequency domain output
\(x[n]\) is the time domain input
\(W_N^k = e^{-j2\pi k/N}\) is the twiddle factor
\(N\) is the number of points (power of 2)
\(k\) is the frequency index
\(n\) is the time index

Goertzel Algorithm

\(y[n] = 2\cos(2\pi f_0/f_s)y[n-1] - y[n-2] + x[n]\)

where:

\(y[n]\) is the filter output
\(x[n]\) is the input signal
\(f_0\) is the frequency of interest
\(f_s\) is the sampling frequency
\(n\) is the time index

Chirp Signal

\(x[n] = \cos(2\pi(f_0n + \frac{\beta}{2}n^2))\)

where:

\(x[n]\) is the chirp signal
\(f_0\) is the starting frequency
\(\beta\) is the rate of frequency change
\(n\) is the time index

Hilbert Transform (Discrete)

\(h[n] = \begin{cases} \frac{2}{\pi n} & n \text{ odd} \\ 0 & n \text{ even} \end{cases}\)

where:

\(h[n]\) is the Hilbert transform impulse response
\(n\) is the time index
\(\pi\) is pi (approximately 3.14159)

Cross-Correlation Coefficient

\(\rho_{xy} = \frac{R_{xy}[0]}{\sqrt{R_{xx}[0]R_{yy}[0]}}\)

where:

\(\rho_{xy}\) is the correlation coefficient
\(R_{xy}[0]\) is the cross-correlation at lag zero
\(R_{xx}[0]\) is the autocorrelation of x at lag zero
\(R_{yy}[0]\) is the autocorrelation of y at lag zero

Discrete Wavelet Transform

\(W[j,k] = \sum_{n} x[n]2^{-j/2}\psi(2^{-j}n - k)\)

where:

\(W[j,k]\) is the wavelet coefficient
\(x[n]\) is the input signal
\(\psi\) is the mother wavelet
\(j\) is the scale parameter
\(k\) is the translation parameter
\(n\) is the time index

Kalman Filter Prediction

\(\hat{x}_{k|k-1} = F_k\hat{x}_{k-1|k-1}\)

where:

\(\hat{x}_{k|k-1}\) is the state prediction
\(F_k\) is the state transition matrix
\(\hat{x}_{k-1|k-1}\) is the previous state estimate
\(k\) is the time step

Kalman Filter Update

\(\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k(z_k - H_k\hat{x}_{k|k-1})\)

where:

\(\hat{x}_{k|k}\) is the updated state estimate
\(K_k\) is the Kalman gain
\(z_k\) is the measurement
\(H_k\) is the measurement matrix
\(\hat{x}_{k|k-1}\) is the state prediction

Autocorrelation Function (Biased)

\(R_{xx}[m] = \frac{1}{N}\sum_{n=0}^{N-|m|-1} x[n]x[n+|m|]\)

where:

\(R_{xx}[m]\) is the autocorrelation function
\(x[n]\) is the signal
\(N\) is the number of samples
\(m\) is the lag
\(n\) is the time index

Levinson-Durbin Recursion

\(a_m^{(m)} = k_m\) \(a_i^{(m)} = a_i^{(m-1)} - k_ma_{m-i}^{(m-1)}\)

where:

\(a_i^{(m)}\) are the reflection coefficients
\(k_m\) is the mth reflection coefficient
\(m\) is the order
\(i\) is the coefficient index

LMS Adaptive Filter

\(w[n+1] = w[n] + \mu e[n]x[n]\)

where:

\(w[n]\) is the filter coefficient vector
\(\mu\) is the step size
\(e[n]\) is the error signal
\(x[n]\) is the input signal vector
\(n\) is the time index

NLMS Adaptive Filter

\(w[n+1] = w[n] + \frac{\mu}{||x[n]||^2}e[n]x[n]\)

where:

\(w[n]\) is the filter coefficient vector
\(\mu\) is the step size
\(e[n]\) is the error signal
\(x[n]\) is the input signal vector
\(||x[n]||^2\) is the squared norm of the input vector

RLS Adaptive Filter

\(w[n] = w[n-1] + k[n]e[n]\)

where:

\(w[n]\) is the filter coefficient vector
\(k[n]\) is the gain vector
\(e[n]\) is the a priori error
\(n\) is the time index

Bandwidth

\(BW = f_H - f_L\)

where:

\(BW\) is the bandwidth
\(f_H\) is the upper cutoff frequency
\(f_L\) is the lower cutoff frequency

Quality Factor

\(Q = \frac{f_c}{BW}\)

where:

\(Q\) is the quality factor
\(f_c\) is the center frequency
\(BW\) is the bandwidth

Polyphase Decomposition

\(H(z) = \sum_{k=0}^{M-1} z^{-k}E_k(z^M)\)

where:

\(H(z)\) is the transfer function
\(E_k(z)\) are the polyphase components
\(M\) is the number of phases
\(k\) is the phase index

Instantaneous Frequency

\(f_i[n] = \frac{1}{2\pi}\frac{d\phi[n]}{dn}\)

where:

\(f_i[n]\) is the instantaneous frequency
\(\phi[n]\) is the instantaneous phase
\(n\) is the time index

Discrete Sine Transform

\(X[k] = \sum_{n=0}^{N-1} x[n]\sin[\frac{\pi(k+1)(n+1)}{N+1}]\)

where:

\(X[k]\) is the transform coefficient
\(x[n]\) is the input signal
\(N\) is the number of samples
\(k\) is the frequency index
\(n\) is the time index

Complex Envelope

\(z[n] = x[n] + j\hat{x}[n]\)

where:

\(z[n]\) is the complex envelope
\(x[n]\) is the original signal
\(\hat{x}[n]\) is the Hilbert transform
\(n\) is the time index

Matched Filter

\(h[n] = s[L-1-n]\)

where:

\(h[n]\) is the filter impulse response
\(s[n]\) is the signal to be detected
\(L\) is the signal length
\(n\) is the time index

Peak Signal-to-Noise Ratio

\(PSNR = 10\log_{10}\left(\frac{MAX^2}{MSE}\right)\)

where:

\(PSNR\) is the peak signal-to-noise ratio
\(MAX\) is the maximum possible signal value
\(MSE\) is the mean squared error

Frequency Response of Moving Average

\(H(e^{j\omega}) = \frac{\sin(\omega M/2)}{M\sin(\omega/2)}e^{-j\omega(M-1)/2}\)

where:

\(H(e^{j\omega})\) is the frequency response
\(M\) is the filter length
\(\omega\) is the angular frequency

Cepstrum

\(c[n] = \mathcal{F}^{-1}\{\log|\mathcal{F}\{x[n]\}|\}\)

where:

\(c[n]\) is the cepstrum
\(x[n]\) is the input signal
\(\mathcal{F}\) represents the Fourier transform
\(\mathcal{F}^{-1}\) represents the inverse Fourier transform
\(n\) is the time index

Linear Prediction

\(\hat{x}[n] = -\sum_{k=1}^{p} a_k x[n-k]\)

where:

\(\hat{x}[n]\) is the predicted sample
\(x[n]\) is the input signal
\(a_k\) are the prediction coefficients
\(p\) is the prediction order
\(k\) is the coefficient index

Equations 50-75

I'll continue with the next 25 common signal processing equations, focusing on more specialized topics and advanced concepts.

Bilinear Transform

\(s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}\)

where:

\(s\) is the continuous-time complex frequency
\(z\) is the discrete-time complex frequency
\(T\) is the sampling period

Yule-Walker Equations

\(R_{xx}[m] = -\sum_{k=1}^p a_k R_{xx}[m-k]\)

where:

\(R_{xx}[m]\) is the autocorrelation sequence
\(a_k\) are the AR coefficients
\(p\) is the model order
\(m\) is the lag
\(k\) is the coefficient index

Zero-Phase Filter

\(H_{zp}(z) = H(z)H(z^{-1})\)

where:

\(H_{zp}(z)\) is the zero-phase filter
\(H(z)\) is the original filter
\(z\) is the complex frequency variable

Gabor Transform

\(X[n,k] = \sum_{m=-\infty}^{\infty} x[m]w[n-m]e^{-j2\pi km/N}\)

where:

\(X[n,k]\) is the time-frequency representation
\(x[m]\) is the input signal
\(w[n]\) is the window function
\(N\) is the number of frequency points
\(n\) is the time index
\(k\) is the frequency index

Bartlett Window

\(w[n] = 1 - |\frac{2n}{N-1} - 1|\)

where:

\(w[n]\) is the window function value
\(n\) is the sample index
\(N\) is the window length

Compression Ratio

\(CR = \frac{n_{bits_{original}}}{n_{bits_{compressed}}}\)

where:

\(CR\) is the compression ratio
\(n_{bits_{original}}\) is the number of bits in original signal
\(n_{bits_{compressed}}\) is the number of bits after compression

Coherence Function

\(\gamma_{xy}^2(f) = \frac{|P_{xy}(f)|^2}{P_{xx}(f)P_{yy}(f)}\)

where:

\(\gamma_{xy}^2(f)\) is the coherence function
\(P_{xy}(f)\) is the cross-spectral density
\(P_{xx}(f)\) and \(P_{yy}(f)\) are power spectral densities
\(f\) is the frequency

Multi-Rate Noble Identity 1

\(H(z^M)↓M = ↓MH(z)\)

where:

\(H(z)\) is the transfer function
\(M\) is the decimation factor
\(↓M\) represents decimation by M

Savitzky-Golay Filter

\(y[n] = \sum_{k=-M}^{M} h[k]x[n+k]\)

where:

\(y[n]\) is the filtered output
\(x[n]\) is the input signal
\(h[k]\) are the Savitzky-Golay coefficients
\(M\) is the half-width of the window
\(n\) is the time index

Phase Delay

\(\tau_p(\omega) = -\frac{\phi(\omega)}{\omega}\)

where:

\(\tau_p(\omega)\) is the phase delay
\(\phi(\omega)\) is the phase response
\(\omega\) is the angular frequency

Kaiser Window

\(w[n] = \frac{I_0(\beta\sqrt{1-((n-\alpha)/\alpha)^2})}{I_0(\beta)}\)

where:

\(w[n]\) is the window function value
\(I_0\) is the modified Bessel function
\(\beta\) is the shape parameter
\(\alpha\) is \((N-1)/2\)
\(N\) is the window length
\(n\) is the sample index

Median Filter

\(y[n] = \text{median}\{x[n-M],...,x[n],...,x[n+M]\}\)

where:

\(y[n]\) is the filtered output
\(x[n]\) is the input signal
\(M\) is the half-width of the window
\(n\) is the time index

Envelope Detection

\(e[n] = |x[n] + j\hat{x}[n]|\)

where:

\(e[n]\) is the envelope
\(x[n]\) is the input signal
\(\hat{x}[n]\) is the Hilbert transform
\(n\) is the time index

Total Harmonic Distortion

\(THD = \sqrt{\frac{\sum_{n=2}^{N} V_n^2}{V_1^2}}\)

where:

\(THD\) is the total harmonic distortion
\(V_n\) is the voltage of the nth harmonic
\(V_1\) is the voltage of the fundamental frequency
\(N\) is the number of harmonics considered

Frequency Modulation

\(x[n] = A\cos(2\pi f_cn + \beta\sin(2\pi f_mn))\)

where:

\(x[n]\) is the FM signal
\(A\) is the amplitude
\(f_c\) is the carrier frequency
\(f_m\) is the modulating frequency
\(\beta\) is the modulation index
\(n\) is the time index

Amplitude Modulation

\(x[n] = A_c(1 + \mu m[n])\cos(2\pi f_cn)\)

where:

\(x[n]\) is the AM signal
\(A_c\) is the carrier amplitude
\(\mu\) is the modulation index
\(m[n]\) is the modulating signal
\(f_c\) is the carrier frequency
\(n\) is the time index

Complementary Filter

\(H_1(z) + H_2(z) = 1\)

where:

\(H_1(z)\) is the first filter transfer function
\(H_2(z)\) is the second filter transfer function
\(z\) is the complex frequency variable

Barker Code Correlation

\(R[k] = \sum_{n=0}^{N-1} b[n]b[n+k]\)

where:

\(R[k]\) is the autocorrelation function
\(b[n]\) is the Barker sequence
\(N\) is the sequence length
\(k\) is the lag
\(n\) is the time index

Notch Filter Transfer Function

\(H(z) = \frac{1 - 2\cos(\omega_0)z^{-1} + z^{-2}}{1 - 2r\cos(\omega_0)z^{-1} + r^2z^{-2}}\)

where:

\(H(z)\) is the transfer function
\(\omega_0\) is the notch frequency
\(r\) is the pole radius
\(z\) is the complex frequency variable

Raised Cosine Filter

\(H(f) = \begin{cases} T, & |f| \leq \frac{1-\alpha}{2T} \\ \frac{T}{2}[1 + \cos(\frac{\pi T}{\alpha}(|f| - \frac{1-\alpha}{2T}))], & \frac{1-\alpha}{2T} \leq |f| \leq \frac{1+\alpha}{2T} \\ 0, & \text{otherwise} \end{cases}\)

where:

\(H(f)\) is the frequency response
\(T\) is the symbol period
\(\alpha\) is the roll-off factor
\(f\) is the frequency

Golay Complementary Sequences

\(G_A(z)G_A(z^{-1}) + G_B(z)G_B(z^{-1}) = 2N\)

where:

\(G_A(z)\) is the z-transform of the first sequence
\(G_B(z)\) is the z-transform of the second sequence
\(N\) is the sequence length
\(z\) is the complex frequency variable

Walsh Transform

\(W[k] = \sum_{n=0}^{N-1} x[n]wal(k,n)\)

where:

\(W[k]\) is the Walsh transform coefficient
\(x[n]\) is the input signal
\(wal(k,n)\) is the Walsh function
\(N\) is the number of points
\(k\) is the sequency index
\(n\) is the time index

Normalized Cross-Correlation

\(\rho[m] = \frac{\sum_{n} (x[n]-\bar{x})(y[n+m]-\bar{y})}{\sqrt{\sum_{n} (x[n]-\bar{x})^2\sum_{n} (y[n]-\bar{y})^2}}\)

where:

\(\rho[m]\) is the normalized correlation
\(x[n]\) and \(y[n]\) are the signals
\(\bar{x}\) and \(\bar{y}\) are the means
\(m\) is the lag
\(n\) is the time index

Zadoff-Chu Sequence

\(x[n] = e^{-j\pi un(n+1)/N}\)

where:

\(x[n]\) is the sequence value
\(u\) is the sequence root
\(N\) is the sequence length
\(n\) is the time index

Maximum Length Sequence

\(s[n] = s[n-p] \oplus s[n-q]\)

where:

\(s[n]\) is the sequence value
\(p\) and \(q\) are the tap positions
\(\oplus\) represents modulo-2 addition
\(n\) is the time index

Equations 75-100

I'll continue with the next 25 equations, focusing on more specialized signal processing concepts and applications.

Multi-Channel Wiener Filter

\(\mathbf{W} = \mathbf{R}_{xx}^{-1}\mathbf{R}_{xy}\)

where:

\(\mathbf{W}\) is the filter coefficient matrix
\(\mathbf{R}_{xx}\) is the input correlation matrix
\(\mathbf{R}_{xy}\) is the cross-correlation matrix

Costas Loop Phase Error

\(e[n] = I[n]Q'[n] - I'[n]Q[n]\)

where:

\(e[n]\) is the phase error signal
\(I[n]\) is the in-phase component
\(Q[n]\) is the quadrature component
\(I'[n]\) and \(Q'[n]\) are the derivatives

Discrete Hartley Transform

\(H[k] = \sum_{n=0}^{N-1} x[n]cas(\frac{2\pi kn}{N})\)

where:

\(H[k]\) is the transform coefficient
\(x[n]\) is the input signal
\(cas(\theta) = \cos(\theta) + \sin(\theta)\)
\(N\) is the transform length
\(k\) is the frequency index

Exponential Moving Average

\(y[n] = \alpha x[n] + (1-\alpha)y[n-1]\)

where:

\(y[n]\) is the filtered output
\(x[n]\) is the input signal
\(\alpha\) is the smoothing factor
\(n\) is the time index

Rician K-Factor

\(K = \frac{A^2}{2\sigma^2}\)

where:

\(K\) is the Rician K-factor
\(A\) is the amplitude of the dominant signal
\(\sigma^2\) is the variance of the multipath components

Periodogram

\(P_{xx}(f) = \frac{1}{N}|X(f)|^2\)

where:

\(P_{xx}(f)\) is the power spectral density
\(X(f)\) is the Fourier transform
\(N\) is the number of samples
\(f\) is the frequency

Teager Energy Operator

\(\Psi[x[n]] = x^2[n] - x[n-1]x[n+1]\)

where:

\(\Psi[x[n]]\) is the Teager energy
\(x[n]\) is the input signal
\(n\) is the time index

Adaptive Line Enhancer

\(y[n] = \mathbf{w}^T[n]\mathbf{x}[n-\Delta]\)

where:

\(y[n]\) is the filter output
\(\mathbf{w}[n]\) is the weight vector
\(\mathbf{x}[n]\) is the input vector
\(\Delta\) is the prediction delay

Modified DFT Filter Bank

\(H_k(z) = H(zW_M^k)\)

where:

\(H_k(z)\) is the kth bandpass filter
\(H(z)\) is the prototype filter
\(W_M = e^{-j2\pi/M}\) is the twiddle factor
\(M\) is the number of channels

Constant Q Transform

\(X[k] = \sum_{n=0}^{N_k-1} x[n]w[n]e^{-j2\pi Qn/N_k}\)

where:

\(X[k]\) is the transform coefficient
\(x[n]\) is the input signal
\(w[n]\) is the window function
\(Q\) is the quality factor
\(N_k\) is the window length for bin k

Short-Time Energy Function

\(E[n] = \sum_{m=-\infty}^{\infty} (x[m]w[n-m])^2\)

where:

\(E[n]\) is the short-time energy
\(x[m]\) is the input signal
\(w[n]\) is the window function
\(n\) is the frame index
\(m\) is the time index

Zero-Crossing Rate

\(ZCR[n] = \sum_{m=-\infty}^{\infty} |\text{sign}(x[m]) - \text{sign}(x[m-1])|w[n-m]\)

where:

\(ZCR[n]\) is the zero-crossing rate
\(x[m]\) is the input signal
\(w[n]\) is the window function
\(\text{sign}()\) is the signum function

Linear Predictive Coding Error

\(e[n] = x[n] - \sum_{k=1}^{p} a_k x[n-k]\)

where:

\(e[n]\) is the prediction error
\(x[n]\) is the input signal
\(a_k\) are the LPC coefficients
\(p\) is the prediction order
\(k\) is the coefficient index

Mel-Frequency Scale

\(f_{mel} = 2595\log_{10}(1 + \frac{f}{700})\)

where:

\(f_{mel}\) is the frequency in mel scale
\(f\) is the frequency in Hz

Bark Scale

\(z = 13\arctan(0.00076f) + 3.5\arctan((f/7500)^2)\)

where:

\(z\) is the frequency in Bark scale
\(f\) is the frequency in Hz

Phase Vocoder

\(\phi_{out}[n] = \phi_{in}[n] + 2\pi f_{in}\frac{R_{out}}{R_{in}}\)

where:

\(\phi_{out}[n]\) is the output phase
\(\phi_{in}[n]\) is the input phase
\(f_{in}\) is the input frequency
\(R_{out}\) is the output rate
\(R_{in}\) is the input rate

Wavelet Packet Transform

\(W_{j,k,n}[m] = 2^{j/2}w_n(2^jm - k)\)

where:

\(W_{j,k,n}[m]\) is the wavelet packet basis
\(w_n\) is the wavelet packet function
\(j\) is the scale parameter
\(k\) is the translation parameter
\(n\) is the oscillation parameter

Preemphasis Filter

\(y[n] = x[n] - \alpha x[n-1]\)

where:

\(y[n]\) is the filtered output
\(x[n]\) is the input signal
\(\alpha\) is the preemphasis coefficient
\(n\) is the time index

Modulation Transfer Function

\(MTF(f) = \frac{|H(f)|}{|H(0)|}\)

where:

\(MTF(f)\) is the modulation transfer function
\(H(f)\) is the system transfer function
\(f\) is the frequency

Karhunen-Loève Transform

\(\Phi_k = \sum_{n=1}^{N} v_k[n]x[n]\)

where:

\(\Phi_k\) is the KLT coefficient
\(v_k[n]\) is the kth eigenvector
\(x[n]\) is the input signal
\(N\) is the signal length

Adaptive Spectral Enhancement

\(H(z) = \frac{1}{1 - \sum_{k=1}^{p} a_k z^{-k}}\)

where:

\(H(z)\) is the enhancement filter
\(a_k\) are the prediction coefficients
\(p\) is the filter order
\(z\) is the complex frequency variable

Pitch Synchronous Overlap Add (PSOLA)

\(y[n] = \sum_{m=-\infty}^{\infty} x[m]h[n-mT_0]\)

where:

\(y[n]\) is the modified signal
\(x[m]\) is the input signal
\(h[n]\) is the synthesis window
\(T_0\) is the pitch period
\(n\) is the time index

Harmonic Plus Noise Model

\(s[n] = h[n] + r[n]\)

where:

\(s[n]\) is the total signal
\(h[n]\) is the harmonic component
\(r[n]\) is the noise residual
\(n\) is the time index

Goertzel Energy

\(E_k = x_1^2[N] + x_2^2[N] - 2\cos(2\pi k/N)x_1[N]x_2[N]\)

where:

\(E_k\) is the energy at frequency k
\(x_1[N]\) and \(x_2[N]\) are Goertzel algorithm outputs
\(N\) is the block length
\(k\) is the frequency bin

Adaptive Noise Cancellation

\(e[n] = d[n] - \mathbf{w}^T[n]\mathbf{x}[n]\)

where:

\(e[n]\) is the error signal
\(d[n]\) is the desired signal
\(\mathbf{w}[n]\) is the filter coefficient vector
\(\mathbf{x}[n]\) is the reference noise vector