Sine waves are important because Fourier´s Theorem states that most signals of practical interest can be decomposed into an infinite sum of sine waves. Discrete-time signals (also called time series) are defined over the set of integers, that is, they are indexed sequences. A discrete-time sine wave is defined by:
Where A is an amplitude and θo is the phase in radians. Meanwhile, ωo=2πf is the angular frequency and x[n] could be written as:
It is important to understand that the frequency of a discrete-time sinusoid is not uniquely defined. This fundamental ambiguity is a consequence of a basic trigonometric property:
In words, the value of a sinusoid does not change if an integer multiple of 2π is added to its argument. Adding the 2πkn to the argument of equation (1) we get:
Two cases must be distinguished. If k≥-f, the equation (2) is equivalent to a sinusoid with frequency f+k with no change in phase:
On the other hand, if k<-f, equation (3) leads to a negative frequency. To avoid this, we introduce:
We also make use of the property:
In consequence, returning to equations (2) and (3), we obtain a sinusoid of frequency l-f with a reversal in phase:
In conclusion, a discrete-time sinusoid with frequency f is identical to a same-phase sinusoid of frequency f+k, where k is any integer greater than –f, or to a phase-reversed sinusoid of frequency l-f if l>f.
Equation (3) can be expressed more concisely using complex exponential notation:
Because value of a complex exponential does not change if a multiple of 2π is added to its argument, we get:
Equation (5) is equivalent to equation (4). Because of this fundamental frequency ambiguity, we will often implicitly assume that the angular frequency of a discrete-time sinusoid is restricted to the range –π≤ω≤π, or equivalent, that -1/2≤f≤1/2.
The Matlab function cos or sin generates the sinusoidal sequences. For example, for x[n]=3cos(0.1πn+π/3)+2sin(0.5πn), 0≤n≤10, we will use the following script:
This script yields:
- Digital Signal Processing Using Matlab, 3erd ed
Literature review by:
Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer
Exercises are solved!!
WhatsApp: +34633129287 Inmediate Attention!!
Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)
Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.
Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs
Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.
Contacto: España. +34633129287
Caracas, Quito, Guayaquil, Jaén.
10 comentarios sobre “The sinusoidal function in discrete time – Matlab”