EECS 216 LECTURE NOTES
CONTINUOUS-TIME AND DISCRETE-TIME SINUSOIDS
DEF: Asinusoidal signal has the standard form x(t) = Acos(ωot+φ).
Where: A≥0=amplitude;ωo≥0=frequency;|φ| ≤ π=phase (shift).
Freq: fo=ωo/(2π)=circular or cyclic frequency in Hertz=cycles per second.
Note: Sinusoid x(t) = Acos(2πfot+φ) has cyclic frequency foHertz.
Period: T=1
fo=2π
ωo→x(t) = x(t+T)for all tusing cos(x) = cos(x+ 2kπ).
DC: DC (constant) signals: ωo=fo= 0; T=∞(low-frequency limit).
RMS(x)= q1
TRT
0A2cos2(ωot+φ)dt =q1
TRT
0
A2
2(1 + cos(2ωot+ 2φ))dt =A
√2
using cos2(x) = 1
2(1 + cos(2x)) and M(cos(t)) = 0 (by inspection).
DC: ωo= 0 →x(t) = Acos φ→RMS(x) = A|cos φ| ≥ 0 (note A≥0).
Note: Can convert other forms to this standard form. Some examples:
1. x(t) = −3 cos(7t+0.1) = 3 cos(7t±π+0.1) using −cos(x) = cos(x±π).
2. x(t) = 3 sin(7t+ 0.1) = 3 cos(7t−π
2+ 0.1) using sin(x) = cos(x−π
2).
3. x(t) = −3 cos(−7t+ 0.1) = 3 cos(7t±π−0.1) using cos(−x) = cos(x).
x(t): Continuous-time sinusoids are periodic with period T=1
fo=2π
ωo.
x[n]: Discrete-time sinusoids are NOT periodic UNLESS fois rational.
ωo= 2πK
T⇔x[n] has period T, provided K
Tis reduced to lowest terms.
Why? Acos(ωon+φ) = Acos(ωo(n+T) + φ)⇔ωoT= 2πk ⇔fo=ωo
2π=k
T.
EX: ωo= 2π→T= 1; ωo= (2.02)π→T= 100; ω0= 2 →nonperiodic.
Freq.: Acos(ωon+φ) = Acos([ωo+ 2kπ]n+φ)→ωoitself is periodic.
As ωoincreases from 0 to π, oscillation rate increases.
As ωoincreases from πto 2π, oscillation rate decreases.
Huh? Frequency range π≤ωo≤2π⇔ −π≤ωo≤0 since ωoperiodic.
WLOG: Restrict discrete-time frequencies to range −π≤ωo≤π.
Note: ωo= 0 →cos(ωon) = 1; ωo=±π→cos(ωon) = (−1)n.
SUM OF MULTIPLE SINUSOIDS
If: x(t) has period Txand freq. fx;y(t) has period Tyand freq. fy;
And: Tx/Tyis a rational number (otherwise x(t) + y(t) not periodic);
Then: x(t) + y(t) has period Tx+y=Least Common Multiple of Txand Ty;
Then: x(t) + y(t) has freq. fx+y=Greatest Common Divisor of fxand fy.
Note: Always works for sinusoids; rarely fails for general periodic signals.
How? (1) Reduce Tx
Ty=M
Nto lowest terms. (2) Tx+y=N Tx=MTy.
Why? Tx
Ty=M
N→x(t+Tx+y) + y(t+Tx+y) = x(t+NTx) + y(t+M Ty).
EX: x(t) = cos(0.3πt) + cos( 2
3πt)→fo=3
20 ,1
3→fx+y=1
60 →Tx+y= 60.
