barcode lib ssrs Thus, the eigenvalue, which we denote by H (elw), is in Software

Draw Code 128B in Software Thus, the eigenvalue, which we denote by H (elw), is

Thus, the eigenvalue, which we denote by H (elw), is
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Note that H(eJw)is, in general, complex-valued and depends on the frequency w of the complex exponential. Thus, it may be written in terms of its real and imaginary parts.
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or in terms of its magnitude and phase,
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where
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FOURIER ANALYSIS
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[CHAP. 2
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(fi(eJ")12 = H(eJ")FZ*(e~") = ~ i ( e j " ) H;(ejo) +
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and Graphical representations of the frequency response are of great value in the analysis of LSI systems, and plots of the magnitude and phase are commonly used. However, another useful graphical representation is a plot of 20 log1H(eJ")I versus o. The units on the log magnitude scale are decibels (abbreviated dB). Thus, 0 dB corresponds to a value of 1H(ejW)l= 1,20dB is equivalent to 1H(ejw)l = 10, -20dB is equivalent to IH(ejU)l = 0.1, and so on. It is also useful to note that 6 dB corresponds approximately to I H (eJ")( = 2, and -6 dB is approxi= mately I H(eJW)l 0.5. One of the advantages of a log magnitude plot is that, because the logarithm expands the scale for small values of (H(ej")(, it is useful in displaying the fine detail of the frequency response near zero. A graphical representation that is often used instead of the phase is the group delay, which is defined as follows:
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In evaluating the group delay, the phase is taken to be a continuous and differentiable function of w by adding integer multiples of n to the principal value of the phase (this is referred to as unwrapping the phase). The function ~ ( e j " )is very useful and important in the characterization of LSI systems and is called the frequency response. The frequency response defines how a complex exponential is changed in (complex) amplitude when it is filtered by the system. The frequency response is particularly useful if we are able to decompose an input signal into a sum of complex exponentials. For example, the response of an LSI system to an input of the form x(n) =
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k= l
ak (ej"~ H )ejnwk
will be
y (n) =
where H(ejw) is the frequency response of the system evaluated at frequency on.
EXAMPLE 2.2.1 Let x ( n ) = cos(n*) be the input to a linear shift-invariant system with a real-valued unit sample response h(n). If x(n) is decomposed into a sum of two complex exponentials,
the response of the system may be written as
Because h(n) is real-valued, H ( e J Wis conjugate symmetric: )
Therefore, and it follows that
CHAP. 21 Periodicity
FOURIER ANALYSIS
The frequency response is a complex-valued function of w and is periodic with a period 23r. This is in sharp contrast with the frequency response of a linear time-invariant continuous-time system, which has a frequency response that is not periodic, in general. The reason for this periodicity stems from the fact that a discrete-time complex exponential of frequency oo is the same as a complex exponential of frequency wo 23r; that is,
Therefore, if the input to a linear shift-invariant system is x(n) = ejnWO, response must be the same as the the response to the signal s ( n ) = eJn(w+2n). This, in turn, requires that
~ ( ~ j " = )f j o
(ej(w0+2")
Symmetry
If h(n) is real-valued, the frequency response is a conjugate symmetric function of frequency:
H (e- j w ) = H *(ej w )
Conjugate symmetry of H(ejW) implies that the real part is an even function of w,
HR(ejW) HR(eCi") =
and that the imaginary part is odd,
Hl(ejW) - ~ [ ( e - j " ) =
Conjugate symmetry also implies that the magnitude is even,
I H (ejW)l= IH (e- j") 1
and that the phase and group delay are odd,
EXAMPLE 2.2.2 Consider the LSI system with unit sample response
a where a is a real number with l1 < I . The frequency response is
The squared magnitude of the frequency response is
and the phase is
H1(eiw)A(@) tan-' - tan-, = HR(ew ) j
-a sin o
-acosw
Finally, the group delay is found by differentiating the phase. The result is
FOURIER ANALYSIS
[CHAP. 2
Inverting the Frequency Response Given the frequency response of a linear shift-invariant system,
the unit sample response may be recovered by integration:
The integral may be taken over any period of length 2n.
EXAMPLE 2.2.3 For a system with a frequency response given by
(this system is referred to as an ideal low-pass filter), the unit sample response is
h(n) = 2~
JWC ,
1 eJ"Wdo = -[eJ""c 2jrcn
sin nu, - e-i""c] = rc n
Note that this system is noncausal (it is also unstable) and, therefore, unrealizable.
23 FILTERS .
The term digitalfilter, or simply filter, is often used to refer to a discrete-time system. A digital filter is defined by J. E Kaiser1 as a ". . . computational process or algorithm by which a sampled signal or sequence of numbers (acting as the input) is transformed into a second sequence of numbers termed the output signal. The computational process may be that of lowpass filtering (smoothing), bandpass filtering, interpolation, the generation of derivatives, etc." Filters may be characterized in terms of their system properties, such as linearity, shift-invariance,causality, stability, etc., and they may be classified in terms of the form of their frequency response. Some of these classifications are described below. Linear Phase
A linear shift-invariant system is said to have linear phase if its frequency response is of the form
~ ( e i " )= A(ejw)e-jaw where ar is a real number and ~ ( e j " ) a real-valued function of w. Note that the phase of H(ei") is is 9h(W)= -a0
i-""
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