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Bandpass Signaling Principles and Circuits Chap. 4
4–5 BANDPASS FILTERING AND LINEAR DISTORTION
Equivalent Low-Pass Filter
In Sec. 2–6, the general transfer function technique was described for the treatment of linear
filter problems. Now a shortcut technique will be developed for modeling a bandpass
filter by using an equivalent low-pass filter that has a complex-valued impulse response.
(See Fig. 4–3a.) v 1 (t) and v 2 (t) are the input and output bandpass waveforms, with the corresponding
complex envelopes g 1 (t) and g 2 (t). The impulse response of the bandpass filter,
h(t), can also be represented by its corresponding complex envelope k(t). In addition, as
shown in Fig. 4–3a, the frequency domain description, H( f), can be expressed in terms of
v 1 (t)=Re [g 1 (t)e j c t ]
Bandpass filter
v 2 (t)=Re [g 2 (t )e j c t ]
h 1 (t) = Re [k 1 (t)e j c t ]
H(f)= –
K(f – f c )+–
K*(–f – f c )
2
2
(a) Bandpass Filter
1
– K*(–f – f c )|
2
|H(f )|
1
– |K(f – f c )|
2
– f c f c
f
(b) Typical Bandpass Filter Frequency Response
–
1 g 1 (t)
2
1
– G 1 (f)
2
Equivalent low-pass filter
–
1
2
k(t)
–
1
2K(f)
–
1 g 2 (t)
2
–
1 G 2 (f)
2
(c) Equivalent (Complex Impulse Response) Low-pass Filter
–
1 |K(f)|
2
f
(d) Typical Equivalent Low-pass Filter Frequency Response
Figure 4–3
Bandpass filtering.