Notes › ENGR 2341: Signals and Systems Lecture 7

Spectrum Signals

calendar_today February 10, 2025 article 216 words access_time 1 min replay Modified April 1, 2025

Table of Contents

Spectrum

The spectrum of a signal is the frequency content of the signal
Represented as a list of frequency-amplitude pairs or a complex amplitude vs frequency plot
- Negative complex amplitudes are impossible! To normalize negative amplitudes, add/subtract $\pi$ to the phase $\phi$

Plots

One-sided spectrum plots use the sum of cosines

$$ \sum_{k=1}^{n}A_{k}\cos(\omega t + \phi_{k}) = A\cos(\omega t+\phi) $$

Two-sided spectrum plots use the Sum of complex exponentials obtained using Inverse Euler

$$ \cos \theta = \frac{e^{j\theta} + e^{-j\theta}}{2} $$

Spectrograms have time on $x$-axis and $\omega$ on $y-axis$, with the magnitude of the spectrum being colored (like a heat map)

Sinusoidal Signals

Signals can be expressed as a sum of sinusoidal signals
Two forms: cosine and complex exponential

Cosine Form

Amplitude is $A$
$\omega$ is radian frequency (rad/s)
$\phi$ is phase (rad)

$$x(t) = A\cos(\omega t+\phi) = A\cos(\omega (t-t_{d}))$$

Complex Exponential form

Each cosine comes with two pieces
- Each piece has half the amplitude
- One piece has $\omega \gt 0$ and the same phase
- The other piece has $\omega \lt 0$ and negated phase

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

References

Spectrum modulated signals (course handout)
Spectrum sinusoidal signals (course handout)
Sinusoidal signal (course handout)

Sources

Sinusoidal signal
Spectrum modulated signals
Spectrum sinusoidal signals

Graph

mindmap ↗