SINUSOIDAL SIGNALS

[Pages:40]SINUSOIDAL SIGNALS

SINUSOIDAL SIGNALS

x(t) = cos(2f t + ) = cos(t + ) (continuous time)

x[n] = cos(2f n + ) = cos(n + ) (discrete time)

f : frequency (s-1 (Hz)) : angular frequency (radians/s) : phase (radians)

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Sinusoidal signals

A

A cos( t) A sin( t)

0

-A

0

T/2

xc(t) = A cos(2f t)

- amplitude A - period T = 1/f - phase: 0

T

3T/2

2T

xs(t) = A sin(2f t) = A cos(2f t - /2)

-amplitude A -period T = 1/f -phase: -/2

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WHY SINUSOIDAL SIGNALS?

? Physical reasons: - harmonic oscillators generate sinusoids, e.g., vibrating structures - waves consist of sinusoidals, e.g., acoustic waves or electromagnetic waves used in wireless transmission

? Psychophysical reason: - speech consists of superposition of sinusoids - human ear detects frequencies - human eye senses light of various frequencies

? Mathematical (and physical) reason: - Linear systems, both physical systems and man-made filters, affect a signal frequency by frequency (hence lowpass, high-pass etc filters)

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EXAMPLE: TRANSMISSION OF A LOW-FREQUENCY SIGNAL USING HIGH-FREQUENCY ELECTROMAGNETIC (RADIO) SIGNAL - A POSSIBLE (CONVENTIONAL) METHOD:

AMPLITUDE MODULATION (AM) Example: low-frequency signal

v(t) = 5 + 2 cos(2ft), f = 20 Hz High-frequency carrier wave

vc(t) = cos(2fct), fc = 200 Hz

Amplitude modulation (AM) of carrier (electromagnetic) wave:

x(t) = v(t) cos(2fct) 5

8 6 4 2 v0 -2 -4 -6 -8

8 6 4 2 x0 -2 -4 -6 -8

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.1

t

Top: v(t) (dashed) and vc(t) = cos(2fct). Bottom: transmitted signal x(t) = v(t) cos(2fct).

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Frequency contents of transmitted signal

x(t) = v(t) cos(2fct) = (5 + 2 cos(2ft)) cos(2fct) = 5 cos(2fct) + 2 cos(2ft) cos(2fct)

Trigonometric identity:

cos

cos

=

1 2

cos(

-

)

+

1 2

cos(

+

)

2 cos(2ft) cos(2fct) = cos (2(fc - f)t)+cos (2(fc + f)t)

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x(t) = 5 cos(2fct) + cos (2(fc - f)t) + cos (2(fc + f)t)

5 Spectrum of v(t)

2

0 20

Spectrum of vc(t)

1

E

200 Frequency

Spectrum of x(t)

5

1

1

E

0

180 200 220

Frequency

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