signal spectra, signal processing

8/3/2019 Signal Spectra, Signal Processing

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Signal Spectra, Signal Processing


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Sinusoidal Signals

Fundamentals building blocks for

describing arbitrary signals.

General signals can be expressed as

sums of sinusoids (Fourier Theory)

Bridge to frequency domain.

Sinusoids are special signals to linear

filters (eigenfunctions)


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Time and Frequency

Closely related via sinusoids

Provide two different perspectives on

signals.

Many operations are easier to understand

in frequency domain.


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Sampling

Conversion from continuous time to

discrete time.

Required for DSP

Converts a signal to a sequence of

numbers (samples).

Straightforward operation.

With a few strange effects.


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Filtering

A simple, but powerful, class of operations

on signals

Filtering transforms an input signal into a

more suitable output signal .

Often best understood in frequency

domain.


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Introduction to Sinusoids

Formula for Sinusoidal Signals

The general formula for a sinusoidal signal is

A, f, and are parameters that characterize the sinusoidalsignal.

Where: A Amplitude: determines the height

of the sinusoid.

f

Frequency: determines the numberof cycles per second.

Phase: determines the location

of the sinusoid.


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Cosine Function

The properties of sinusoidal signals stem from theproperties of the cosine function:

Periodicity: cos(x+2) = cos(x)

Eveness: cos(-x) = cos(x)

Ones of cosine: cos(2k) = 1, for all integers k. Minus ones of cosine: cos((2k+1)) = -1, for all integers

k.

Zeros of cosine: cos(/2(2k+1)) = 0, for all integers k.

Relationship to sine function:sin(x) = cos(x /2) (shift to the right)

cos(x) = sin(x + /2) (shift to the left)


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Amplitude

The amplitude A is a scaling factor.

It determines how large the signal is

Specifically, the sinusoid oscillates

between +A and

A.


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Frequency and Period

Sinusoids are periodic signals

The frequency f indicates how many times

the sinusoid repeats per second

The duration of each cycle is called the

period of the sinusoid and denoted by T.

The relationship between frequency and

period is f = 1/T and T = 1/f


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Phase and Delay

The phase,, causes a sinusoid to be shiftedsideways

A sinusoid with phase, = 0, has a maximum at t =

0. A sinusoid that has a maximum at t = t1 can be

written as x(t) = Acos(2f(t t1))

Expanding the argument of the cosine leads tox(t) = Acos(2ft

2ft1)

Comparing to the general formula for a sinusoidreveals = -2ft1 and t1 = -/2 f


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Phase and Delay


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Example 1

Convert sine function to cosine functionand vice versa.

(a) x1(t) = 8sin(40t + /8)

(b) x2(t) = 6cos(70t - /9)

(c) x3(t) = 5sin(20t + 3/2)

(d) x4(t) = 10cos(30t + /4)


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Solution

(a) x1(t) = 8cos(40t + /8 - /2)

= 8cos(40t - 3 /8)

(b) x2(t) = 6sin(70t - /9 + /2)

= 6sin(70t + 7/18)

(c) x3(t) = 5cos(20t + 3/2 - /2)

= 5cos(20t + )

(d) x4(t) = 10sin(30t + /4 + /2)

= 10sin(30t + 3/4)


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Example 2

Find the equation that will represent afunction, given in the figure.


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Solution

Amplitude = 2

Period = 0.22 0.02 = 0.42 0.22 = 0.2 = T;

f = 1/T = 1/0.2 = 5 Hz

Time delay = 0.02 = (-)2(5)(0.02) = (-)/5

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

0.02) = 2cos(10t/5)

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