Digital filters - Continuous and Discrete Time Signals and Systems - page 631

Number Phase

of taps, N z-transfer function, H ( z ) (linear or non-linear) Phase value

− 2 z − 1 + 2 z − 2 − z − 3 − 1 . 5 Ω + π/ 2

type 4, linear

− z − 2 type 3, linear − Ω + π/ 2

+ 2 z − 1 + 2 z − 2 non-linear

+ 2 z − 1 + 2 z − 2 + z − 3 type 2, linear − 1 . 5 Ω

+ 2 z − 1 − 2 z − 2 + z − 3 non-linear

+ 2 z − 1 + 3 z − 2 + 2 z − 3 + z − 4 type 1, linear − 2 Ω

+ 2 z − 1 + 3 z − 2 + 2 z − 3 − z − 4 non-linear

( N − 1)
2

Substituting m = − k in the above equation, we obtain

+ ( N − 1) / 2

N − 1

2

N − 1

2

− j N − 1) Ω / 2

H ( Ω ) = e h 2 h − m cos( m Ω )

m − 1

+ ( N − 1) / 2

N − 1

2

N − 1

2

− j( N − 1) Ω / 2

= e h 2 h − k cos( k Ω )

k = 1

( N − 1) / 2

− j( N − 1) Ω / 2

= e a [ k ] cos( k Ω ) ,

k = 0

where a [0] = h [( N − 1) / 2] and a [ k ] = 2 h [( N − 1) / 2 − k ]. It is observed that
the derived H ( Ω ) matches with Eq. (14.9), with α = ( N

− 1) / 2 and G ( Ω )given

in Table 14.2.

Example 14.1
Determine if the FIR filters specified in column 2 of Table 14.3 have linear
phase or not. Also determine the value of the phase.

Solution
The phase linearity can be determined using the conditions given in Eq. (14.8).
The third column of Table 14.3 shows whether a filter is linear phase and the
type of linear-phase filter. The phase function, i.e. ( −α Ω

+ β ) in Eq. (14.9), is

shown in the fourth column.
To confirm the results of the last two entries of Table 14.3, Fig. 14.5 plots
the magnitude and phase spectra of the FIR filter specified in the second to last
row of Table 14.3. The phase plot in Fig. 14.5(b) confirms that the FIR filter
has a linear phase. Since a phase of π is the same as that of −π , the sharp

= 0.5 π are not discontinuities but correspond to the same
value. The magnitude spectrum illustrates non-uniform gains within the pass
band and stop bands, implying that the FIR filter is not an ideal lowpass filter
despite having a linear phase.

transitions at Ω