Sinusoidal Signal

Introduction

The sinusoidal signal is the cornerstone of signal processing and the most fundamental periodic waveform. When Joseph Fourier showed in the early 1800s that any periodic signal can be decomposed into a sum of sinusoids, he gave us a universal tool for understanding signals. Every sound you hear is a combination of sinusoids (harmonics). Every radio signal is a sinusoid (carrier wave). The response of any LTI system to a sinusoidal input is another sinusoid at the same frequency — making sinusoids the natural basis for frequency-domain analysis.

For B.Tech students, mastering sinusoidal signals means understanding frequency, phase, amplitude, and their interplay. This single topic connects to Fourier series, Fourier transforms, phasors in circuit analysis, modulation in communications, and virtually every topic in this course.

Mathematical Definition

Continuous-Time Sinusoid

The general continuous-time sinusoidal signal is:

$$x(t) = A\cos(\omega_0 t + \phi)$$

where:

• $A$ = amplitude (peak value, always positive)
• $\omega_0 = 2\pi f_0$ = angular frequency in rad/s
• $f_0$ = cyclic frequency in Hz (cycles per second)
• $T_0 = 1/f_0 = 2\pi/\omega_0$ = fundamental period in seconds
• $\phi$ = phase angle in radians (determines where the cosine "starts")

Equivalently using sine: $x(t) = A\sin(\omega_0 t + \phi + \pi/2)$, since $\cos\theta = \sin(\theta + \pi/2)$.

Physical Interpretation

Phase $\phi$ represents a time shift: $A\cos(\omega_0 t + \phi) = A\cos(\omega_0(t + \phi/\omega_0))$. A positive phase $\phi > 0$ shifts the waveform to the left (earlier in time) by $\phi/\omega_0$ seconds.

The period $T_0$ is the time for one complete oscillation cycle. The frequency $f_0$ tells us how many cycles occur per second.

Complex Exponential Representation

Using Euler's formula, the cosine can be written as:

$$A\cos(\omega_0 t + \phi) = \frac{A}{2}e^{j(\omega_0 t + \phi)} + \frac{A}{2}e^{-j(\omega_0 t + \phi)}$$

$$= \frac{Ae^{j\phi}}{2}e^{j\omega_0 t} + \frac{Ae^{-j\phi}}{2}e^{-j\omega_0 t}$$

This reveals that a real cosine consists of two complex exponentials: one at positive frequency $+\omega_0$ and one at negative frequency $-\omega_0$. This is why the spectrum of a real sinusoid has two lines (positive and negative frequency components).

Phasor Representation

In steady-state AC circuit analysis, we represent a sinusoid by its complex amplitude (phasor):

$$A\cos(\omega_0 t + \phi) \quad \leftrightarrow \quad \tilde{X} = Ae^{j\phi}$$

The phasor captures amplitude and phase while the frequency is implicit (assumed known). This simplifies circuit analysis enormously — differential equations become algebraic equations.

Fourier Transform of a Sinusoid

A cosine signal $x(t) = A\cos(\omega_0 t + \phi)$ that exists for all time has the Fourier transform:

$$X(\omega) = \pi A e^{j\phi}\delta(\omega - \omega_0) + \pi A e^{-j\phi}\delta(\omega + \omega_0)$$

Or in terms of cyclic frequency $f$:

$$X(f) = \frac{A}{2}e^{j\phi}\delta(f - f_0) + \frac{A}{2}e^{-j\phi}\delta(f + f_0)$$

This means a sinusoid has zero bandwidth — all its energy is concentrated at a single frequency. This is an idealization (physically, any signal has finite duration and therefore nonzero bandwidth), but it's a powerful conceptual tool.

Properties of Sinusoidal Signals

Periodicity

A continuous-time sinusoid $A\cos(\omega_0 t + \phi)$ is always periodic with period $T_0 = 2\pi/\omega_0$ (assuming $\omega_0 \neq 0$).

Power Signal

A sinusoid has infinite energy but finite average power:

$$P = \lim_{T \to \infty}\frac{1}{2T}\int_{-T}^{T}|A\cos(\omega_0 t + \phi)|^2 dt = \frac{A^2}{2}$$

So the RMS (root-mean-square) value is $A_{rms} = A/\sqrt{2}$. This is why AC voltage is quoted as RMS — a 220V RMS sine wave has peak amplitude $220\sqrt{2} \approx 311$ V.

Orthogonality

Sinusoids at different frequencies are orthogonal over one period:

$$\int_0^{T_0} \cos(m\omega_0 t)\cos(n\omega_0 t) \, dt = \begin{cases} T_0/2 & m = n \neq 0 \\ T_0 & m = n = 0 \\ 0 & m \neq n \end{cases}$$

This orthogonality is the mathematical foundation of Fourier series — it allows us to extract individual frequency components by computing inner products (correlation with sinusoids at each frequency).

LTI System Response

When a sinusoid $A\cos(\omega_0 t + \phi)$ is input to an LTI system with frequency response $H(j\omega)$:

$$\text{Output} = A|H(j\omega_0)|\cos(\omega_0 t + \phi + \angle H(j\omega_0))$$

The frequency stays the same! Only the amplitude is scaled by $|H(j\omega_0)|$ and the phase is shifted by $\angle H(j\omega_0)$. This eigensignal property is the foundation of frequency-domain analysis.

Discrete-Time Sinusoid

The discrete-time sinusoid is:

$$x[n] = A\cos(\omega_0 n + \phi)$$

where $\omega_0$ is the digital frequency in radians per sample.

Key Differences from Continuous-Time

1. Periodicity condition: $x[n]$ is periodic only if $\omega_0/(2\pi)$ is a rational number. If $\omega_0 = 2\pi(M/N)$ where $M/N$ is in lowest terms, the period is $N$ samples.

1. Frequency uniqueness: Frequencies that differ by $2\pi$ are indistinguishable: $\cos((\omega_0 + 2\pi)n) = \cos(\omega_0 n)$. So only the range $0 \leq \omega_0 < 2\pi$ (or $-\pi < \omega_0 \leq \pi$) contains unique frequencies.

1. Highest frequency: The fastest oscillation occurs at $\omega_0 = \pi$ (alternating $+1, -1, +1, -1, \ldots$). Beyond $\pi$, frequencies actually become "slower" due to aliasing.

Harmonic Relationships

When multiple sinusoids are added:

$$x(t) = \sum_{k=1}^{N} A_k \cos(k\omega_0 t + \phi_k)$$

the result is periodic with period $T_0 = 2\pi/\omega_0$. The frequency $f_0$ is the fundamental, and $kf_0$ are the harmonics. This is exactly the structure of the Fourier series — any periodic signal can be written in this form.

Worked Examples

Example 1: Find the period and frequency of $x(t) = 3\cos(100\pi t + \pi/4)$.

Solution: $\omega_0 = 100\pi$ rad/s, so $f_0 = \omega_0/(2\pi) = 50$ Hz and $T_0 = 1/50 = 0.02$ s = 20 ms.

Example 2: Is $x[n] = \cos(3n)$ periodic?

Solution: We need $3/(2\pi) = 3/(2\pi)$ to be rational. Since $3/(2\pi)$ is irrational, $x[n]$ is NOT periodic.

Example 3: A sinusoid $5\cos(2000\pi t)$ passes through a system with $H(j\omega)|_{\omega=2000\pi} = 0.5e^{-j\pi/3}$. Find the output.

Solution: Output $= 5 \times 0.5 \cos(2000\pi t - \pi/3) = 2.5\cos(2000\pi t - \pi/3)$.

Example 4: Find the average power of $x(t) = 3\cos(10t) + 4\sin(10t)$.

Combine: $x(t) = 5\cos(10t - \arctan(4/3))$ (amplitude = $\sqrt{9+16} = 5$)

Power: $P = 5^2/2 = 12.5$ watts.

Applications

• AC power systems: Mains voltage is a 50/60 Hz sinusoid
• Radio/TV broadcasting: Carrier waves are sinusoids modulated by information
• Music/acoustics: Pure tones are sinusoids; complex sounds are sums of harmonics
• Vibration analysis: Mechanical resonances produce sinusoidal oscillations
• Testing LTI systems: Frequency response is measured by sweeping a sinusoidal input across frequencies

Key Takeaways

• A sinusoid is characterized by three parameters: amplitude $A$, frequency $\omega_0$, and phase $\phi$
• The Fourier transform of a sinusoid is a pair of impulses at $\pm\omega_0$ — zero bandwidth
• Average power is $A^2/2$; RMS value is $A/\sqrt{2}$
• LTI systems preserve the frequency of a sinusoidal input, changing only amplitude and phase
• Discrete-time sinusoids are periodic only when $\omega_0/(2\pi)$ is rational
• Discrete-time frequencies are unique only in $(-\pi, \pi]$ due to the $2\pi$ periodicity
• Orthogonality of sinusoids at different frequencies enables Fourier decomposition