Sinusoids

Sinusoids

The sinusoid is the fundamental building block of signal processing. Every real-world signal — audio, radio, seismic — can be decomposed into a sum of sinusoids by the Fourier theorem.

The Sinusoidal Signal

A continuous sinusoidal signal is described by:

$x(t) = A \sin(2\pi f t + \phi)$

where:

• $A$ = amplitude (peak value)
• $f$ = frequency in Hz (cycles per second)
• $t$ = time in seconds
• $\phi$ = phase offset in radians

Angular Frequency

It is often convenient to write the angular frequency:

$\omega = 2\pi f \quad \text{(radians per second)}$

so the signal becomes $x(t) = A \sin(\omega t + \phi)$.

Example

A 440 Hz sinusoid (concert A) with amplitude 1 and zero phase:

$x(t) = \sin(2\pi \cdot 440 \cdot t)$

At $t = 1/1760$ seconds (quarter cycle), $x = \sin(\pi/2) = 1.0$ — the peak.

Your Task

Implement:

• sinusoid(A, f, t, phi=0) — returns $A \sin(2\pi f t + \phi)$
• angular_frequency(f) — returns $\omega = 2\pi f$

import math

def sinusoid(A, f, t, phi=0):
    return A * math.sin(2 * math.pi * f * t + phi)

def angular_frequency(f):
    return 2 * math.pi * f

# 440 Hz sine at t=0 (zero crossing)
print(round(sinusoid(1.0, 440.0, 0.0), 4))   # 0.0
# At quarter period: t = 1/(4*440)
print(round(sinusoid(1.0, 440.0, 1/1760), 4)) # 1.0
print(round(angular_frequency(60.0), 4))       # 376.9911