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:
n=[0:10]; x=3*cos(0.1*pi*n+pi/3)+2*sin(0.5*pi*n);
stem(n,x)
This script yields:
Source:
- Digital Signal Processing Using Matlab, 3erd ed
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