Signal Processing Mathematics

Table of Contents

1. Continuous vs. Discrete Signals
2. Nyquist-Shannon Sampling Theorem
3. Aliasing
4. Fourier Series
5. Discrete Fourier Transform (DFT)
6. Fast Fourier Transform (FFT)
7. Z-Transform
8. Laplace Transform
9. Convolution Theorem
10. FIR vs. IIR Filters
11. Windowing
12. Frequency Response & Phase Delay
13. Power Spectral Density
14. Wiener Filter
15. Wavelets
16. FAQ

1. Continuous vs. Discrete Signals

A continuous-time signal x(t) is defined for every real value of time t. It is the natural description of physical quantities — voltage across a capacitor, acoustic pressure, temperature — that vary smoothly over time. A discrete-time signal x[n] is defined only at integer sample indices n. It arises whenever a continuous signal is measured at regular intervals: x[n] = x(nT), where T is the sampling period and fs = 1/T is the sampling rate in hertz.

Energy and power give two useful signal categories. An energy signal has finite total energy:

E = integral from -inf to +inf of |x(t)|^2 dt  <  infinity

For discrete-time:
E = sum from n=-inf to +inf of |x[n]|^2  <  infinity

A power signal has finite average power but infinite energy (e.g., a sinusoid that lasts forever). Most periodic signals are power signals. Noise processes are commonly analyzed as power signals characterized by their power spectral density.

Signal Classifications

2. Nyquist-Shannon Sampling Theorem

The central theorem of digital signal processing: a bandlimited continuous-time signal with maximum frequency component fmax can be perfectly reconstructed from uniformly spaced samples taken at rate fs, provided:

fs > 2 * fmax

Nyquist rate fN = 2 * fmax
Nyquist frequency fN/2 = fmax = fs / 2

The reconstruction formula uses sinc interpolation:

x(t) = sum from n=-inf to +inf of x[n] * sinc( (t - n*T) / T )

where sinc(u) = sin(pi*u) / (pi*u)
and T = 1/fs is the sampling period

In the frequency domain, sampling creates a periodic replication of X(f) with period fs. When fs is large enough, the replicas do not overlap and a lowpass reconstruction filter with cutoff at fs/2 recovers the original spectrum exactly.

Worked Example: Audio Sampling

fmax = 20,000 Hz
Minimum fs = 2 * 20,000 = 40,000 Hz = 40 kHz

CD standard uses fs = 44,100 Hz
This provides a safety margin above 40 kHz
and accommodates the transition band of the
anti-aliasing lowpass filter.

Anti-Aliasing Filter

Before sampling any real-world signal, an anti-aliasing lowpass filter must be applied to remove all frequency content above fs/2. Without this filter, high-frequency components above the Nyquist frequency fold back into the baseband and corrupt the sampled signal irreversibly. The anti-aliasing filter is the analog lowpass filter whose cutoff is set at or below fs/2 — it is a necessary hardware component in every ADC front end.

3. Aliasing

Aliasing occurs when a signal is sampled below its Nyquist rate. The spectral replicas created by sampling overlap, and frequency components above fs/2 appear to the sampled system as lower frequencies. The alias frequency of a tone at frequency f when sampled at rate fs is:

f_alias = | f - round(f/fs) * fs |

Equivalently, frequencies fold around fs/2:
f_alias = | f mod fs - fs/2 | is not quite right — use:

For f in range (fs/2, fs):
  f_alias = fs - f   (folds down from fs)

For f in range (fs, 3*fs/2):
  f_alias = f - fs   (same as baseband)

Worked Example: Aliasing a 1200 Hz Tone

Sampling rate:  fs = 1000 Hz
Signal tone:    f  = 1200 Hz   (above Nyquist frequency 500 Hz)

Alias:  f_alias = 1200 - 1000 = 200 Hz

The 1200 Hz tone is indistinguishable from
a 200 Hz tone after sampling at 1000 Hz.
Once aliased, the original frequency
cannot be recovered.

Temporal aliasing is visible in video: a spinning propeller or wagon wheel filmed at 24 fps can appear to spin backward when its rotation rate places a blade in a position that matches a lower apparent rotation rate. The same phenomenon in audio causes heterodyne interference in undersampled audio ADCs.

4. Fourier Series

Any periodic function x(t) with period T0 satisfying the Dirichlet conditions can be expressed as a Fourier series. The trigonometric form is:

x(t) = a0/2 + sum from n=1 to inf of
         [ an * cos(n * omega0 * t) + bn * sin(n * omega0 * t) ]

omega0 = 2*pi / T0   (fundamental angular frequency)

Coefficients:
  a0 = (2/T0) * integral over one period of x(t) dt
  an = (2/T0) * integral over one period of x(t)*cos(n*omega0*t) dt
  bn = (2/T0) * integral over one period of x(t)*sin(n*omega0*t) dt

Complex Exponential Form

Using Euler's formula, the compact complex exponential form is:

x(t) = sum from n=-inf to +inf of cn * exp(j*n*omega0*t)

cn = (1/T0) * integral from 0 to T0 of x(t)*exp(-j*n*omega0*t) dt

Relationship to real coefficients:
  c0  = a0/2
  cn  = (an - j*bn) / 2   for n > 0
  c-n = cn*              (complex conjugate)

Gibbs Phenomenon

When the Fourier series of a function with a jump discontinuity is truncated to N terms, the partial sum overshoots the function value near the discontinuity. The overshoot converges to approximately 8.9% of the jump height — the Gibbs constant of roughly 9% — regardless of how many terms are included. The width of the overshoot narrows as N increases but the peak amplitude does not decrease. In signal processing this manifests as ringing artifacts when a rectangular filter (brick-wall bandpass) is applied. Windowing functions and smooth filter roll-offs reduce the Gibbs effect at the cost of slightly reduced frequency resolution.

Parseval's Theorem for Fourier Series

(1/T0) * integral over one period of |x(t)|^2 dt
  = sum from n=-inf to +inf of |cn|^2
  = (a0/2)^2 + (1/2)*sum from n=1 to inf of (an^2 + bn^2)

Interpretation: total average power = sum of power in each harmonic

Parseval's theorem is frequently used to evaluate the sums of infinite series. For example, the Fourier series of the square wave provides the identity: 1 + 1/9 + 1/25 + ... = pi^2/8 (sum of 1/(2k-1)^2 for k=1 to infinity).

5. Discrete Fourier Transform (DFT)

The DFT converts a finite sequence of N time-domain samples x[0], x[1], ..., x[N-1] into N complex frequency-domain values X[0], X[1], ..., X[N-1]:

DFT:
  X[k] = sum from n=0 to N-1 of x[n] * exp(-j * 2*pi * k * n / N)
       = sum from n=0 to N-1 of x[n] * W_N^(k*n)

where W_N = exp(-j * 2*pi / N)  is the twiddle factor base

Inverse DFT (IDFT):
  x[n] = (1/N) * sum from k=0 to N-1 of X[k] * exp(j * 2*pi * k * n / N)

Frequency Resolution and Bin Interpretation

The DFT bin index k corresponds to physical frequency f_k = k * fs / N. The frequency resolution (bin spacing) is Delta_f = fs / N. To achieve 1 Hz resolution with a 44,100 Hz sample rate, one needs N = 44,100 samples. The first N/2 bins correspond to positive frequencies 0 to fs/2 - Delta_f; bins N/2 through N-1 correspond to negative frequencies.

DFT Properties

Worked Example: 4-Point DFT

Sequence: x[n] = [1, 1, 1, 1],  N = 4
W_4 = exp(-j*pi/2) = -j

X[0] = x[0]*W4^0 + x[1]*W4^0 + x[2]*W4^0 + x[3]*W4^0
     = 1 + 1 + 1 + 1 = 4

X[1] = x[0]*1 + x[1]*(-j) + x[2]*(-1) + x[3]*(j)
     = 1 - j - 1 + j = 0

X[2] = 1 + (-1) + 1 + (-1) = 0

X[3] = 1 + j + (-1) + (-j) = 0

Result: X[k] = [4, 0, 0, 0]
Interpretation: DC component only (as expected for a constant signal).

6. Fast Fourier Transform (FFT)

The Cooley-Tukey radix-2 decimation-in-time (DIT) FFT splits an N-point DFT (N = 2^m) recursively into two N/2-point DFTs of the even-indexed and odd-indexed samples:

X[k] = X_even[k] + W_N^k * X_odd[k]       for k = 0, ..., N/2 - 1
X[k + N/2] = X_even[k] - W_N^k * X_odd[k]   (butterfly structure)

where:
  X_even[k] = DFT of x[0], x[2], x[4], ..., x[N-2]
  X_odd[k]  = DFT of x[1], x[3], x[5], ..., x[N-1]
  W_N^k = exp(-j*2*pi*k/N)  (twiddle factor)

Butterfly Diagram (8-Point FFT)

An 8-point FFT requires 3 stages (log2(8) = 3). Each stage performs N/2 = 4 butterfly operations. The input samples are bit-reversed before the first stage.

Bit-reversal permutation for N=8:
  Index:     0   1   2   3   4   5   6   7
  Binary:  000 001 010 011 100 101 110 111
  Reversed:000 100 010 110 001 101 011 111
  New idx:   0   4   2   6   1   5   3   7

Stage 1 (W_8^0 = 1, W_8^2, W_8^4 = -1, ...):
  Butterfly pairs: (x[0],x[1]), (x[2],x[3]), (x[4],x[5]), (x[6],x[7])
  Each butterfly:
    a' = a + W * b
    b' = a - W * b

Stage 2: span-2 butterflies
Stage 3: span-4 butterflies → final X[k]

Complexity: (N/2)*log2(N) complex multiplications
  N=1024:  512 * 10 = 5,120  vs  1,048,576 for direct DFT

Computational Savings

7. Z-Transform

The Z-transform is the discrete-time counterpart of the Laplace transform. For a sequence x[n], the bilateral Z-transform is:

X(z) = sum from n=-infinity to +infinity of x[n] * z^(-n)

z is a complex variable: z = r * exp(j*omega)

One-sided (unilateral) Z-transform:
X(z) = sum from n=0 to +infinity of x[n] * z^(-n)
(used when x[n] = 0 for n < 0, i.e., causal sequences)

Region of Convergence (ROC)

The ROC is the set of z values for which the sum converges absolutely. The ROC is always a connected annular region r1 < |z| < r2 in the complex plane.

Right-sided (causal) sequence:
  ROC: |z| > r_max  (exterior of circle)
  Includes z = infinity

Left-sided (anticausal) sequence:
  ROC: |z| < r_min  (interior of circle)
  Includes z = 0

Two-sided sequence:
  ROC: r1 < |z| < r2  (annular ring, if non-empty)

BIBO stability condition:
  System is stable iff ROC includes the unit circle |z| = 1

Common Z-Transform Pairs

Inverse Z-Transform

Three main methods for the inverse Z-transform:

Transfer Function of a Discrete System

For a linear time-invariant (LTI) discrete system described by the difference equation:

sum from k=0 to N of a[k]*y[n-k] = sum from k=0 to M of b[k]*x[n-k]

Taking Z-transform and solving for H(z) = Y(z)/X(z):

H(z) = (b[0] + b[1]*z^-1 + ... + b[M]*z^-M)
       / (a[0] + a[1]*z^-1 + ... + a[N]*z^-N)

H(z) = B(z) / A(z)   (ratio of polynomials in z^-1)

Poles: roots of A(z) = 0
Zeros: roots of B(z) = 0

Stable system: all poles inside unit circle |z| < 1

8. Laplace Transform & Transfer Functions

The Laplace transform is the continuous-time analog of the Z-transform. For a signal x(t), the bilateral Laplace transform is:

X(s) = integral from -inf to +inf of x(t) * exp(-s*t) dt

s = sigma + j*omega   (complex frequency)

One-sided (unilateral):
X(s) = integral from 0 to +inf of x(t) * exp(-s*t) dt

The Fourier transform is the special case s = j*omega (sigma = 0). The Laplace transform extends this to complex s, allowing the analysis of growing or decaying exponential signals.

Common Laplace Transform Pairs

Transfer Function H(s)

For a continuous LTI system with input X(s) and output Y(s), the transfer function is H(s) = Y(s) / X(s). For a system described by a linear ODE with constant coefficients, H(s) is always a rational function of s:

H(s) = (b_M * s^M + b_(M-1)*s^(M-1) + ... + b_0)
       / (a_N * s^N + a_(N-1)*s^(N-1) + ... + a_0)

Poles: values of s where H(s) -> infinity  (denominator = 0)
Zeros: values of s where H(s) = 0         (numerator = 0)

BIBO stability: all poles must have Re(s) < 0
(all poles in left half of complex s-plane)

Frequency response:  H(j*omega) = H(s)|_(s = j*omega)
                     (evaluate on imaginary axis)

Worked Example: First-Order RC Lowpass Filter

RC circuit differential equation:
  RC * dy/dt + y(t) = x(t)

Laplace transform (zero initial conditions):
  s*RC*Y(s) + Y(s) = X(s)

Transfer function:
  H(s) = Y(s)/X(s) = 1 / (1 + s*RC)
       = (1/RC) / (s + 1/RC)

Pole at s = -1/RC  (left half plane -> stable)
Cutoff frequency: omega_c = 1/RC rad/s, f_c = 1/(2*pi*RC) Hz

|H(j*omega)| = 1 / sqrt(1 + (omega*RC)^2)
At omega = 1/RC:  |H| = 1/sqrt(2) ≈ -3 dB  (3 dB point)

9. Convolution Theorem

The output y(t) of any LTI system is the convolution of the input x(t) with the system's impulse response h(t):

y(t) = x(t) * h(t) = integral from -inf to +inf of x(tau)*h(t-tau) dtau

Discrete-time:
y[n] = x[n] * h[n] = sum from k=-inf to +inf of x[k]*h[n-k]

Convolution theorem:
  Y(omega) = X(omega) * H(omega)   (pointwise multiplication)

The output spectrum equals the input spectrum
multiplied by the frequency response H(omega).

Fast Convolution via FFT

Direct convolution of an N-sample signal with an M-sample filter requires O(N*M) operations. Using the FFT convolution theorem, the same result is computed as:

1. Zero-pad both sequences to length L >= N + M - 1
2. Compute X[k] = FFT(x, L)          O(L log L)
3. Compute H[k] = FFT(h, L)          O(L log L)
4. Multiply Y[k] = X[k] * H[k]       O(L)
5. Recover y[n] = IFFT(Y, L)         O(L log L)
Total: O(L log L) vs O(N*M)

Note: zero-padding to L >= N+M-1 converts circular
convolution (DFT property) into linear convolution.

Correlation vs. Convolution

Cross-correlation measures the similarity between two signals as a function of time lag:

Cross-correlation:
  R_xy(tau) = integral of x*(t) * y(t + tau) dt
            = x*(-t) convolved with y(t)

Autocorrelation:
  R_xx(tau) = integral of x*(t) * x(t + tau) dt

Key difference from convolution: no time-reversal of x.
Correlation detects signal presence in noise (matched filter);
convolution implements filtering.

10. FIR vs. IIR Filters

FIR Filter

A Finite Impulse Response (FIR) filter of order M is described by the difference equation:

y[n] = sum from k=0 to M of b[k] * x[n-k]

Transfer function:
  H(z) = sum from k=0 to M of b[k] * z^(-k)
       = B(z)   (numerator polynomial only, no denominator)

All poles at z = 0  -> always stable
Impulse response: h[n] = b[n] (the coefficients themselves)

FIR filters with symmetric coefficients (b[k] = b[M-k]) have exactly linear phase, meaning all frequencies are delayed by the same constant M/2 samples. This preserves waveform shape and is essential in applications like ECG processing and audio effects.

IIR Filter

An Infinite Impulse Response (IIR) filter uses feedback. The general difference equation is:

y[n] = sum from k=0 to M of b[k]*x[n-k]
     - sum from k=1 to N of a[k]*y[n-k]

Transfer function:
  H(z) = B(z) / A(z)
       = (b0 + b1*z^-1 + ... + bM*z^-M)
         / (1 + a1*z^-1 + ... + aN*z^-N)

Stable iff all poles are inside unit circle.

Classic IIR Filter Designs

FIR vs. IIR Comparison

11. Windowing

Windowing multiplies a finite data record by a smoothly tapered function before computing the DFT to reduce spectral leakage. The main trade-off is between main lobe width (frequency resolution) and sidelobe level (dynamic range / leakage rejection).

Common Window Functions (length N)

Rectangular:  w[n] = 1                                 (no windowing)
  Sidelobe level: -13 dB   Main lobe width: 4*pi/N

Hann (von Hann):
  w[n] = 0.5 * (1 - cos(2*pi*n / (N-1)))
  Sidelobe level: -31 dB   Main lobe width: 8*pi/N

Hamming:
  w[n] = 0.54 - 0.46 * cos(2*pi*n / (N-1))
  Sidelobe level: -42 dB   Main lobe width: 8*pi/N

Blackman:
  w[n] = 0.42 - 0.5*cos(2*pi*n/(N-1)) + 0.08*cos(4*pi*n/(N-1))
  Sidelobe level: -74 dB   Main lobe width: 12*pi/N

Blackman-Harris:
  4-term cosine window
  Sidelobe level: -92 dB   Main lobe width: 16*pi/N

Flat-top:
  Near-unity gain across main lobe; best for amplitude accuracy
  Sidelobe level: -93 dB   Very wide main lobe

Window Selection Guidelines

12. Frequency Response, Phase Delay & Group Delay

The frequency response H(e^(j*omega)) of a discrete system is obtained by evaluating the Z-transform transfer function H(z) on the unit circle z = e^(j*omega):

H(e^(j*omega)) = H(z)|_(z = e^(j*omega))

Magnitude response:  |H(e^(j*omega))|
Phase response:      angle(H(e^(j*omega))) = theta(omega)

Phase delay:
  tau_p(omega) = -theta(omega) / omega
  Time delay experienced by a sinusoid at frequency omega.

Group delay:
  tau_g(omega) = -d(theta(omega)) / d(omega)
  Delay experienced by the envelope of a narrowband signal.
  Constant group delay = linear phase = all frequencies delayed equally.

Linear phase (constant group delay) is essential in applications where waveform shape must be preserved: audio, ECG, radar pulse compression. FIR filters with symmetric coefficients achieve exactly linear phase. IIR filters can approximate constant group delay using all-pass equalizer cascades.

Bode Plot Fundamentals

The Bode plot displays |H(j*omega)| in decibels (20*log10|H|) and the phase angle in degrees versus log frequency. Key asymptotic rules for a pole at s = -p (or zero at s = -z):

Simple pole at s = -p  (p > 0):
  Magnitude: 0 dB for omega << p, slopes at -20 dB/decade for omega >> p
  Phase:     0 degrees for omega << p/10, -90 degrees for omega >> 10*p
             crosses -45 degrees at omega = p

Simple zero at s = -z  (z > 0):
  Magnitude: +20 dB/decade after omega = z
  Phase:     +90 degrees contribution

Double pole: -40 dB/decade slope, -180 degrees phase
n poles at origin: -20n dB/decade, -90n degrees at all frequencies

13. Power Spectral Density

Power spectral density (PSD) characterizes how power is distributed across frequencies in a random or stationary signal. For a wide-sense stationary (WSS) process:

Autocorrelation:
  R_xx(tau) = E[ x(t) * x*(t + tau) ]

Wiener-Khinchin theorem:
  S_xx(f) = F( R_xx(tau) )
           = integral from -inf to +inf of R_xx(tau) * exp(-j*2*pi*f*tau) dtau

Total power:
  P = R_xx(0) = integral from -inf to +inf of S_xx(f) df

Units: S_xx(f) in units of [signal^2 / Hz]  (e.g., V^2/Hz)

PSD Estimation Methods

Common Noise PSDs

White noise:        S_xx(f) = N0 / 2 = constant   (flat spectrum)
Pink (1/f) noise:   S_xx(f) = C / f               (electronics, music, finance)
Brownian noise:     S_xx(f) = C / f^2             (random walk / integration of white noise)

For a system H(s) driven by white noise with PSD N0/2:
  Output PSD:  S_yy(f) = |H(j*2*pi*f)|^2 * N0/2

14. Wiener Filter

The Wiener filter is the optimal linear filter for signal estimation in the presence of additive noise, in the minimum mean-square error (MMSE) sense. Given an observed signal y[n] = s[n] + v[n] (desired signal s plus noise v), the Wiener filter estimates s[n] from y[n].

Wiener-Hopf equation (in frequency domain):
  H_opt(f) = S_sy(f) / S_yy(f)

For uncorrelated signal and noise:
  S_yy(f) = S_ss(f) + S_vv(f)
  S_sy(f) = S_ss(f)   (cross-PSD of desired signal with observation)

Therefore:
  H_opt(f) = S_ss(f) / (S_ss(f) + S_vv(f))

When SNR(f) = S_ss(f) / S_vv(f):
  H_opt(f) = SNR(f) / (1 + SNR(f))

At high SNR: H_opt(f) -> 1  (pass everything)
At low SNR:  H_opt(f) -> 0  (attenuate noise-dominated frequencies)

The Wiener filter requires knowledge of the signal and noise PSDs (S_ss and S_vv). When these are unknown, they must be estimated from data. The Kalman filter is the time-domain, recursive, non-stationary generalization of the Wiener filter.

Wiener Filter Applications

• Audio denoising: suppress background noise between speech bursts
• Image restoration: remove blur and noise from photographs (2D Wiener filter)
• Radar: extract target reflection from clutter and thermal noise
• Channel equalization: compensate for frequency-selective fading in communications
• Biomedical: remove EMG artifact from ECG recordings

15. Wavelets

Unlike the Fourier transform, which uses infinitely long sinusoids with no time localization, the wavelet transform uses short, oscillatory functions (wavelets) that are scaled and shifted to provide simultaneous time-frequency localization. This is especially valuable for non-stationary signals whose frequency content changes over time.

Continuous Wavelet Transform (CWT)

CWT: W(a, b) = (1/sqrt(|a|)) * integral from -inf to +inf of
               x(t) * psi*( (t-b)/a ) dt

a = scale parameter   (a > 0; large a -> low frequency / stretched)
b = translation parameter (time localization)
psi(t) = mother wavelet
psi*  = complex conjugate of psi

Admissibility condition (required for reconstruction):
  C_psi = integral of |PSI(f)|^2 / |f| df  <  infinity
  Requires PSI(0) = 0  (zero mean: wavelet has no DC component)

Discrete Wavelet Transform (DWT)

The DWT samples the CWT on a dyadic grid: a = 2^j, b = k * 2^j for integers j and k. This is implemented efficiently using a two-channel filter bank (Mallat's algorithm):

At each decomposition level j:

  Approximation coefficients:
    cA_j[k] = sum_n x[n] * h[n - 2k]   (lowpass filter + downsample by 2)

  Detail coefficients:
    cD_j[k] = sum_n x[n] * g[n - 2k]   (highpass filter + downsample by 2)

h[n] = scaling filter (lowpass)   — derived from scaling function phi
g[n] = wavelet filter (highpass)  — derived from mother wavelet psi

For orthogonal wavelets:  g[n] = (-1)^n * h[L-1-n]  (quadrature mirror filters)

Reconstruction (synthesis):
  x[n] = sum_k cA_j[k]*h[n-2k] + sum_k cD_j[k]*g[n-2k]  (upsample + filter)

Multiresolution Analysis (MRA)

Mallat's multiresolution analysis framework provides the mathematical foundation for the DWT. An MRA is a sequence of nested closed subspaces V_j of L^2(R) satisfying:

... subset V_2 subset V_1 subset V_0 subset V_-1 subset V_-2 subset ...

Closure under scaling: f(t) in V_j <=> f(2t) in V_(j-1)
Union dense in L^2(R),  Intersection = (0)

V_(j-1) = V_j  (+)  W_j    (direct sum: W_j are detail spaces)

DWT decomposes signal at each level:
Level 0: original signal
Level 1: cA1 (approx 0-fs/4) + cD1 (detail fs/4-fs/2)
Level 2: cA2 (0-fs/8) + cD2 (fs/8-fs/4) + cD1 (fs/4-fs/2)
...continuing until desired resolution

Haar Wavelet

The Haar wavelet is the simplest orthogonal wavelet. Its mother wavelet and scaling function are:

Scaling function (father wavelet):
  phi(t) = 1   for 0 <= t < 1
  phi(t) = 0   otherwise

Mother wavelet:
  psi(t) = +1  for 0 <= t < 0.5
  psi(t) = -1  for 0.5 <= t < 1
  psi(t) =  0  otherwise

Haar filter coefficients:
  h = [1/sqrt(2),  1/sqrt(2)]   (lowpass / scaling)
  g = [1/sqrt(2), -1/sqrt(2)]   (highpass / wavelet)

Haar DWT of [4, 6, 8, 2]:
  Level 1 lowpass (avg pairs):  [5*sqrt(2), 5*sqrt(2)]  -> approx: [5, 5]
  Level 1 highpass (diff pairs): [-sqrt(2), 3*sqrt(2)]  -> detail: [-1, 3]

Common Wavelet Families

Wavelet Applications

• Image compression: JPEG 2000 uses DWT instead of DCT (superior for medical imaging)
• Signal denoising: threshold detail coefficients (hard/soft thresholding)
• EEG/ECG analysis: detect transient events localized in time and frequency
• Seismology: identify P-wave and S-wave arrivals in earthquake records
• Turbulence analysis: study energy cascade across scales
• Finance: detect regime changes and cycles in time series at multiple scales

Frequently Asked Questions