Digital filters - Continuous and Discrete Time Signals and Systems - page 628

frequency Ω = H ( Ω ) = 1 ,< H ( Ω ) =∓ m 1 Ω 1 ;

Ω 1

frequency Ω = H ( Ω ) = 1 ,< H ( Ω ) =∓ m 2 Ω 2 ;

Ω 2

frequency Ω = H ( Ω ) = 0 ,< H ( Ω ) =∓ m 3 Ω 3 ;

Ω 3

where m 1 , m 2 , and m 3 are the slopes of the phase response at Ω =

Ω 1 , Ω 2 and

Ω 3 , respectively.
Using the convolution property, the DTFT of the output of the filter is given

by

Ω 1 )e j m 1 Ω + δ ( Ω

− j m 1 Ω ]

Y ( Ω ) = A 1 π [ δ ( Ω

− −

Ω 1 )e

Ω 2 )e j m 2 Ω + δ ( Ω

− j m 2 Ω ]

+ A 2 π [ δ ( Ω + −

Ω 2 )e

+ A 3 π [ δ ( Ω +

Ω 3 ) + δ ( Ω

−

Ω 3 )] 0 .

Taking the inverse DTFT of the above equation, we obtain

y [ k ] = A 1 cos( Ω 1 ( k − m 1 )) + A 2 cos( Ω 2 ( k − m 2 )) .

For m 1 = m 2 , the input tones A 1 cos( Ω 1 k ) and A 2 cos( Ω 2 k ) are delayed une-
qually and the output sequence y [ k ] is a distorted version of the sinusoidal
components present within the pass band of the filter. To retain the shape of the
pass-band components, each sinusoidal term A 1 cos( Ω 1 k ) and A 2 cos( Ω 2 k )in
y [ k ] should be delayed equally, i.e. m 1 = m 2 . In signal processing, the following

two types of delays are defined:

phase delay d p =−φ ( Ω ) / Ω ;

=− d φ ( Ω )
d Ω

roup delay d g ;

where φ ( Ω ) is the phase of the filter transfer function, i.e. φ ( ω ) = H ( ω ).
In other words, the phase delay ( d p ) is defined as the phase divided by the
frequency, whereas the group delay ( d g ) is defined as the derivative of the phase
with respect to frequency. From the above definitions, it is observed that the
delay of a filter will be constant if the phase φ ( Ω ) of the filter is a linear function
of frequency. A filter is said to have a linear phase response if it satisfies the
following relationships.

φ ( Ω ) =−α Ω , or φ ( Ω ) =−α Ω + β.

The first condition ensures that the filter has constant phase and group delay,
whereas the second condition ensures only constant group delay. Although it is
desirable to have both constant group and phase delays, a constant group delay
is generally sufficient in many applications.
Based on the above discussion, the conditions for distortionless filtering,
where the pass-band components are retained precisely at the filter output, are
enlisted as follows.