Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 353

filter is considered next, followed by the lowpass to bandpass, and lowpass to
bandstop transformations.

7.4.1 Lowpass to highpass filter

The transformation that converts a lowpass filter with the transfer function H ( S )
into a highpass filter with transfer function H ( s )isgivenby

= ξ p
s

S , (7.67)

where S = σ + j ω represents the lowpass domain and s = γ + j ξ represents
the highpass domain. The frequency ξ = ξ p represents the pass-band corner
frequency for the highpass filter. In terms of the CTFT domain, Eq. (7.67) can
be expressed as follows:

ω =− ξ p
ξ

=− ξ p or ξ

ω .

(7.68)

Figure 7.12 shows the effect of applying the frequency transformation in
Eq. (7.68) to the specifications of a highpass filter. Equation (7.68) maps the
highpass specifications in the range −∞ <ξ ≤ 0 to the specifications of a
lowpass filter in the range 0 ≤ ω<∞ . Similarly, the highpass specifications
for the positive range of frequencies (0 <ξ ≤∞ ) are mapped to the lowpass
specifications within the range −∞ ≤ ω< 0. Since the magnitude spectra are
symmetrical about the y -axis, the change from positive ξ frequencies to negative
ω frequencies does not affect the nature of the filter in the entire domain.

Highpass to lowpass transformation
w = − x p / x

w w

| H lp ( w )|

stop band

x p

x s

transition band

1

pass band

x

d s

1− d p

1+ d p

1+ d p

H hp ( x )

1− d p

pass band transition

band

stop band

d s

Fig. 7.12. Highpass to lowpass
transformation.

x

− x p − x s

0