Total energy E and average power P on a per-ohm basis are in VS .NET

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Total energy E and average power P on a per-ohm basis are
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i 2 ( t ) d t joules i 2 ( t ) dt watts
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For an arbitrary continuous-time signal x(t), the normalized energy content E of x ( t ) is defined as
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The normalized average power P of x ( t ) is defined as
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Similarly, for a discrete-time signal x[n], the normalized energy content E of x[n] is defined as
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SIGNALS AND SYSTEMS
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[CHAP. 1
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The normalized average power P of x[n] is defined as 1 P = lim N + - 2 N + 1 ,,= N Based on definitions (1.14) to (1.17), the following classes of signals are defined: 1. x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E < m, and so P = 0. 2. x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P < m, thus implying that E = m. 3. Signals that satisfy neither property are referred to as neither energy signals nor power signals. Note that a periodic signal is a power signal if its energy content per period is finite, and then the average power of this signal need only be calculated over a period (Prob. 1.18).
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1.3 BASIC CONTINUOUS-TIME SIGNALS
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A. The Unit Step Function: The unit step function u(t), also known as the Heaciside unit function, is defined as
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which is shown in Fig. 1-4(a). Note that it is discontinuous at t = 0 and that the value at t = 0 is undefined. Similarly, the shifted unit step function u(t - to) is defined as
which is shown in Fig. 1-4(b).
Fig. 1-4 ( a ) Unit step function; ( b )shifted unit step function.
B. The Unit Impulse Function: The unit impulse function 6(t), also known as the Dirac delta function, plays a central role in system analysis. Traditionally, 6(t) is often defined as the limit of a suitably chosen conventional function having unity area over an infinitesimal time interval as shown in
CHAP. 11
SIGNALS AND SYSTEMS
Fig. 1-5
Fig. 1-5 and possesses the following properties:
But an ordinary function which is everywhere 0 except at a single point must have the integral 0 (in the Riemann integral sense). Thus, S(t) cannot be an ordinary function and mathematically it is defined by
where 4 ( t ) is any regular function continuous at t An alternative definition of S(t) is given by
= 0.
Note that Eq. (1.20) or (1.21) is a symbolic expression and should not be considered an ordinary Riemann integral. In this sense, S(t) is often called a generalized function and 4 ( t ) is known as a testing function. A different class of testing functions will define a different generalized function (Prob. 1.24). Similarly, the delayed delta function 6(t - I,) is defined by
4 ( t ) W - to) dt
=4Po)
(1.22)
where 4 ( t ) is any regular function continuous at t = to. For convenience, S(t) and 6 ( t - to) are depicted graphically as shown in Fig. 1-6.
SIGNALS AND SYSTEMS
[CHAP. 1
Fig. 1-6 ( a ) Unit impulse function; ( b )shifted unit impulse function.
Some additional properties of S ( t ) are
S(- t ) =S(t)
x ( t ) S ( t ) = x(O)S(t)
if x ( t ) is continuous at
t = 0.
x ( t ) S ( t - t o ) = x ( t o ) 6 ( t- t , )
if x ( t ) is continuous at t = to. Using Eqs. (1.22) and ( 1.241, any continuous-time signal x(t can be expressec
Generalized Derivatives:
If g( t ) is a generalized function, its nth generalized derivative g("Y t ) = dng(t )/dt " is defined by the following relation:
where 4 ( t ) is a testing function which can be differentiated an arbitrary number of times and vanishes outside some fixed interval and @ " ' ( t ) is the nth derivative of 4(t).Thus, by Eqs. ( 1.28) and (1.20) the derivative of S( t ) can be defined as
where 4 ( t ) is a testing function which is continuous at t = 0 and vanishes outside some fixed interval and $ ( 0 ) = d 4 ( t ) / d t l , = o . Using Eq. (1.28), the derivative of u ( t ) can be shown to be S ( t ) (Prob. 1.28); that is,
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