NeuralBlitz Signal Processing
What I Do
I provide comprehensive signal processing tools including Fourier analysis, filter design, sampling theory, spectral estimation, and adaptive filtering for engineering and data science applications.
When to Use Me
- Fourier transform analysis
- Filter design and implementation
- Spectral analysis
- Sampling and reconstruction
- Noise reduction
- Adaptive filtering
Core Concepts
- Fourier Analysis: DFT, FFT, DTFT
- Filter Design: IIR, FIR, window methods
- Sampling Theory: Nyquist, aliasing, reconstruction
- Spectral Estimation: Periodogram, Welch, AR
- Adaptive Filtering: LMS, RLS, Kalman
- Multirate: Decimation, interpolation
- Wavelets: DWT, time-frequency analysis
- Statistical Signal: Detection, estimation
Code Examples
Fourier Analysis
import numpy as np
from scipy.fft import fft, ifft, fftfreq
def discrete_fourier_transform(x):
N = len(x)
X = np.zeros(N, dtype=complex)
for k in range(N):
for n in range(N):
X[k] += x[n] * np.exp(-2j * np.pi * k * n / N)
return X
def fast_fourier_transform(x):
return fft(x)
def inverse_dft(X):
N = len(X)
x = np.zeros(N)
for n in range(N):
for k in range(N):
x[n] += X[k] * np.exp(2j * np.pi * k * n / N)
return x / N
def spectral_magnitude(X):
return np.abs(X)
def spectral_phase(X):
return np.angle(X)
def frequency_resolution(fs, N):
return fs / N
x = np.array([1, 2, 3, 4, 5])
X = fast_fourier_transform(x)
freqs = fftfreq(len(x), 1/1000)
print(f"Frequencies: {freqs[:len(freqs)//2]}")
Filter Design
from scipy.signal import firwin, butter, lfilter, freqz
def lowpass_fir_design(N, cutoff, fs, window='hamming'):
nyquist = fs / 2
taps = firwin(N, cutoff/nyquist, window=window)
return taps
def butterworth_lowpass(order, cutoff, fs):
nyquist = fs / 2
b, a = butter(order, cutoff/nyquist, btype='low')
return b, a
def iir_design(order, wp, ws, gpass, gstop, ftype='ellip'):
b, a = ellip(order, gpass, gstop, [wp/nyquist, ws/nyquist])
return b, a
def window_function(window, N):
windows = {
'rectangular': np.ones(N),
'hanning': np.hanning(N),
'hamming': np.hamming(N),
'blackman': np.blackman(N)
}
return windows.get(window, np.ones(N))
def fir_filter_coefficients(N, cutoff, fs, response='lowpass'):
nyquist = fs / 2
if response == 'lowpass':
return firwin(N, cutoff/nyquist)
elif response == 'highpass':
return firwin(N, cutoff/nyquist, pass_zero=False)
N = 64
fs = 1000
cutoff = 100
taps = lowpass_fir_design(N, cutoff, fs)
print(f"Filter order: {N}, Cutoff: {cutoff} Hz")
Sampling Theory
def nyquist_rate(signal_bandwidth):
return 2 * signal_bandwidth
def aliasing_check(f_signal, f_sampling):
if f_signal > f_sampling / 2:
aliased_freq = abs(f_signal - f_sampling)
return True, aliased_freq
return False, f_signal
def anti_aliasing_filter_design(f_pass, f_stop, f_sampling):
f_nyquist = f_sampling / 2
return f_pass / f_nyquist, f_stop / f_nyquist
def sample_and_hold(hold_time, aperture_error):
return hold_time, aperture_error
def reconstruction_sinc(x, t, Ts):
n = np.arange(len(x))
t_grid, n_grid = np.meshgrid(t, n)
sinc = np.sinc(t_grid/Ts - n_grid)
return np.dot(x, sinc)
def delta_sigma_modulation(quantizer_levels, oversampling_ratio):
return quantizer_levels / oversampling_ratio
f_signal = 400
f_sampling = 1000
is_aliased, f_aliased = aliasing_check(f_signal, f_sampling)
print(f"Aliased: {is_aliased}, Frequency: {f_aliased} Hz")
Spectral Estimation
from scipy.signal import welch, periodogram
def periodogram_spectrum(x, fs):
f, Pxx = periodogram(x, fs=fs)
return f, Pxx
def welch_psd(x, fs, nperseg=256):
f, Pxx = welch(x, fs=fs, nperseg=nperseg)
return f, Pxx
def bartlett_method(x, L, fs):
N = len(x)
K = N // L
P_bartlett = np.zeros(L)
for k in range(K):
x_k = x[k*L:(k+1)*L]
P_bartlett += np.abs(fft(x_k, L))**2
return P_bartlett / K
def ar_spectrum(y, order, fs):
from scipy.signal import lfilter
a = arburg(y, order)[0]
w, H = freqz(1, a, worN=1024, fs=fs)
P_ar = np.abs(H)**2
return w, P_ar
def coherence_function(x, y, fs):
from scipy.signal import csd
f, Pxx = welch(x, fs=fs)
f, Pyy = welch(y, fs=fs)
f, Pxy = csd(x, y, fs=fs)
Cxy = np.abs(Pxy)**2 / (Pxx * Pyy)
return f, Cxy
def cepstrum_analysis(x):
log_spectrum = np.log(np.abs(fft(x)))
cepstrum = np.real(ifft(log_spectrum))
return cepstrum
fs = 1000
f, Pxx = welch(x, fs=fs)
print(f"Peak frequency: {f[np.argmax(Pxx)]:.1f} Hz")
Adaptive Filtering
def lms_filter(x, d, mu, order):
N = len(x)
w = np.zeros(order)
y = np.zeros(N)
e = np.zeros(N)
for n in range(order, N):
X = x[n-order:n][::-1]
y[n] = np.dot(w, X)
e[n] = d[n] - y[n]
w = w + 2 * mu * e[n] * X
return y, e, w
def rls_filter(x, d, lambda_, order, delta=1.0):
N = len(x)
w = np.zeros(order)
P = np.eye(order) / delta
y = np.zeros(N)
e = np.zeros(N)
for n in range(order, N):
X = x[n-order:n][::-1]
y[n] = np.dot(w, X)
e[n] = d[n] - y[n]
K = P @ X / (lambda_ + X @ P @ X)
w = w + K * e[n]
P = (P - np.outer(K, X @ P)) / lambda_
return y, e, w
def nlms_filter(x, d, mu, order, epsilon=1e-6):
N = len(x)
w = np.zeros(order)
y = np.zeros(N)
e = np.zeros(N)
for n in range(order, N):
X = x[n-order:n][::-1]
norm = np.linalg.norm(X) + epsilon
y[n] = np.dot(w, X)
e[n] = d[n] - y[n]
w = w + (mu / norm**2) * e[n] * X
return y, e, w
def affine_projection_filter(x, d, mu, order, K):
N = len(x)
w = np.zeros(order)
y = np.zeros(N)
e = np.zeros(N)
for n in range(order, N, K):
X = np.array([x[n-i-order:n-i][::-1] for i in range(K)])
Y = X @ w
E = d[n:n+K] - Y
w = w + mu * X.T @ np.linalg.inv(X @ X.T + 1e-6 * np.eye(K)) @ E
return y, e, w
Best Practices
- Windowing: Choose appropriate windows
- Zero Padding: For interpolation, not resolution
- Leakage: Minimize with proper windowing
- Numerical: Use stable filter structures
- Quantization: Consider fixed-point effects
Common Patterns
# Hilbert transform
def hilbert_transform(x):
return np.imag(hilbert(x))
Core Competencies
- Fourier analysis
- Filter design
- Spectral estimation
- Sampling theory
- Adaptive filtering