Signals and Systems - Electrical Engineering

396 C H A P T E R 6: Application to Control and Communications

Remarks

n According to Equation (6.43) when either

n The transition band is narrowed (i.e.,p→s), or

n The lossαminis increased, or

n The lossαmaxis decreased

the quality of the filter is improved at the cost of having to implement a filter with a high order N.

n The minimum order N is an integer larger or equal to the right side of Equation (6.43). Any integer

larger than the minimum N also satisfies the specifications but increases the complexity of the filter.

n Although there is a range of possible values for the half-power frequency, it is typical to make the frequency

response coincide with either the passband or the stopband specifications giving a value for the half-power

frequency in the range. Thus, we can have either

hp=

p

( 10 0.1αmax− 1 )^1 /^2 N

(6.44)

or

hp=

s

( 10 0.1αmin− 1 )^1 /^2 N

(6.45)

as possible values for the half-power frequency.

n The design aspect is clearly seen in the flexibility given by the equations. We can select out of an infinite

possible set of values of N and of half-power frequencies. The optimal order is the smallest value of N and

the half-power frequency can be taken as one of the extreme values.

n After the factorization, or the formation of D(S)from the poles, we need to denormalize the obtained

transfer function HN(S)= 1 /D(S)by letting S=s/hpto get HN(s)= 1 /D(s/hp), the filter that

satisfies the specifications. If the desired DC gain is not unit, the filter needs to be denormalized in

magnitude by multiplying it by an appropriate gain K.

6.5.3 Chebyshev Low-Pass Filter Design

The normalized magnitude-squared function for the Chebyshev low-pass filter is given by

|HN(′)|^2 =

1

1 +ε^2 C^2 N(/p)

′=



p

(6.46)

where the frequency is normalized with respect to the passband frequencypso that′=/p,

Nstands for the order of the filter,εis a ripple factor, andCN(.)are the Chebyshev orthogonal^4

polynomials of the first kind defined as

CN(′)=

{

cos(Ncos−^1 (′)) |′|≤ 1

cosh(Ncosh−^1 (′)) |′|> 1

(6.47)

The definition of the Chebyshev polynomials depends on the value of′. Indeed, whenever|′|>1,

the definition based in the cosine is not possible since the inverse would not exist; thus the cosh(.)

(^4) Pafnuty Chebyshev (1821–1894), a brilliant Russian mathematician, was probably the first one to recognize the general concept of
orthogonal polynomials.