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Notes › ENGR 2341: Signals and Systems Lecture 11

Spectrum

calendar_today February 18, 2025 article 124 words access_time 1 min replay Modified February 18, 2025

Table of Contents

Sum of Sinusoids

Can be determined by inspection
A list of frequency-complex amplitude pairs
- For one-sided spectrum, use Sum of cosines
- For two-sided spectrum, use Sum of complex exponentials

Using Fourier Series

Possible to use Fourier series representation if the signal is not a sum of sinusoids
There will be an infinite number of terms in the spectrum corresponding to the Fourier series

$$X(j\omega) = \left\{(\omega_{k}, a_{k})\right\}, k \in (-\infty, \infty)$$

Using Fourier Transform

The spectrum of any continuous-time signal is its Fourier transform
It is expressed as a continuous function of frequency, $\omega$

$$X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} \ \mathrm{d}t$$

Using Fourier Analysis

Only useful for a piecewise periodic function
Determine $a_0, \omega_0, a_k$ to find the infinite spectrum

References

Spectrum (course handout)

Sources

Spectrum

Graph

mindmap ↗