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Unformatted text preview: r H a (s). T herefore, ha (t)
i s t he inverse Laplace transform o f H a (s) o r t he inverse Fourier t ransform o f
Ha{jw). T hus, ha{t) 1 = 27r j "IT
.
Ha(jw)e Jwt dw
"IT (12.81a) Recall t hat a d igital filter frequency response is periodic w ith t he first period in
t he frequency range ~w<
Hence, t he b est we could hope is t o realize
t he equivalence o f Ha{jw) w ithin t his r ange. For t his r eason, t he l imits of
i ntegration a re t aken from  7r I T t o 7r I T. T herefore, according t o Eq. (12.80) ¥ h[k] ¥. = T ha(kT) = T
2
7r j "IT Ha(jw)e iwkT dw
"IT T his f requency response is a n a pproximation o f t he desired frequency response
Ha(jw) b ecause of t he t runcation o f h[k]. T hus, Ha(jw) ~ L h[k]eiWkT
k How do we select h[k] for t he b est a pproximation i n t he sense of minimizing t he
energy of t he e rror Ha{jw)  H[e iwT ]? We have a lready solved t his p roblem i n
Sec. 3.32. T he above e quation shows t hat t he r ighthand side is t he f inite t erm
e xponential Fourier series for Ha(jw) w ith p eriod 27r I T. As seen from Eq. (12.81b),
h[k] a re t he F ourier coefficients. We also know t hat a finite Fourier series is t he
o ptimum ( in t he sense of minimizing t he e rror energy) for a given No a ccording t o
t he finality p roperty o f t he F ourier coefficients discussed in Sec. 3.32. t Clearly, t his
choice of h[k] is o ptimum in t he sense of t he m inimum m ean s quared error. For t he
obvious reason, t his m ethod is also known as t he F ourier series method.
• (12.81b) 2. Windowing
F or linear phase filters, we generally s tart w ith zero phase filters for which
H a(jw) is e ither real o r imaginary. T he i mpulse response ha (t) is either a n even
o r o dd f unction of t (see P rob. 4.11). I n e ither case, h a(t) is centered a t t = 0
a nd h as infinite d uration in general. B ut h[k] m ust have only a finite d uration
a nd i t m ust s tart a t k = 0 (causal) for filter realizability. Consequently, t he
h[k] found in s tep 1 needs t o b e t runcated u sing a n N opoint window a nd t hen E xample 1 2.10
Design an ideal lowpass filter for audio band with cutoff frequency 20 kHz. Use
a sixthorder nonrecursive filter using rectangular and Hamming windows. The highest
frequency to be processed is F h = 40 kHz.
In this case n = 6 and No = n + 1 = 7. First we shall choose a suitable value for T .
According to Eq. (12.58)
T<~
 Wh = _ 1_ = 12.5 x 1 0 6
2 Fh tNote that this finite term Fourier series corresponds to the rectangular window function. For
windows other than rectangular, the optimality does not hold. 7 64 12 F requency R esponse a nd D igital F ilters 12.8 Recall t hat a c ontinuoustime sinusoid o f frequency w, d uring d igital processing a ppears
as a d iscretetime sinusoid o f f requency !1 = w T. T he c utoff frequency W e = 27r(20, 000) =
40,00()7r a ppears as a d iscretetime sinusoid of frequency !1 e = W eT = 40, 0007r(12.5 X 1 0 6 ) = i, a nd We 2~ sine ideal ~ ..... . = 2;' T he d esired (zero phas...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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