Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 89

H L ( f )

1

LPF

f, Hz

−f c 0 f c

−f s −f s /2 f s /2 f s

H H ( f )

1

HPF

f, Hz

f s /2−f c f s /2+f c

0

−f s f s

−f s /2 f s /2

H BP ( f )

← f o =(f 1 +f 2 )/2

1

BPF

f, Hz

−f 2 −f 1 f 1 f 2

0

−f s −f s /2 f s /2 f s

H BS ( f )

1

BSF

f, Hz

−f 2 −f 1 0 f 1 f 2

−f s −f s /2 f s /2 f s

Fig. 2.15 Transfer functions of ideal digital filters

Digital Filter Transformations in the Time-Domain

Similar definitions to that of the LPF can be created for HPF, BPF, and BSF filters,
These definitions are provided below, and illustrative plots for the corresponding
transfer functions are provided in Fig. ( 2.15 ).

1. LP ! HP : from Fig. ( 2.15 ) it is seen that the transfer function of a HPF with
cutoff frequency f c is a spectrally-shifted version of a LPF with the same cutoff
frequency, i.e.,

H H ð f Þ¼ H L ð f f s = 2 Þ;

X

h H ð n Þ e j2pnfT s ¼ X

1 1

h L ð n Þ e j2pn ð f þ f s = 2 Þ T s

ð 2 : 20 Þ

n ¼1 n ¼1

¼ X

1

h L ð n Þð 1 Þ n e j2pnfT s

n ¼1

noting that e -j2p n (f s /2)T s = e -jnp = (- 1) n . A simple examination of the above

equation indicates that:

h H ð n Þ¼ð 1 Þ n h L ð n Þ:

i.e., the impulse response of a highpass filter can be obtained from a lowpass one
by simply inverting every second sample.
2. LP ? BP: from Fig. ( 2.15 ) it is apparent that: