Bilinear z transformaion

BILINEAR Z-TRANSFORMAION
AND
PASCAL’S TRIANGLE

NGUYEN SI PHUOC – VUT
SIPHUOCNGUYEN@GMAIL.C
OM

INTRODUCTION
• Bilinear z-transformation is the most common method for
converting the transfer function H(s) of the analog filter to the
transfer function H(z) of the digital filter and vice versa. In this
work, introducing the relationship between the digital
coefficients and the analog coefficients in the matrix equation
definitely involves the Pascal’s triangle.

BILINEAR Z-TRANSFORM
•

The effective and popular
method is currently used to
convert an analog filter in sdomain into an equivalent digital
filter in z-domain is the bilinear ztransform. This technique is one
to one mapping the poles and
zeros on the left half stable
region in s-plane into inside unit
circle in z-plane as shown in fig.1

THE MAIN ADVANTAGE AND
DISADVANTAGE OF BILINEAR ZTRANSFORM
The main advantage of this method is
transform a stable designed analog filter to
a stable digital filter which the frequency
response has the same characteristics as
frequency response of the analog filter.
However, this method will give a non-linear
relationship between analog frequency ωA
and digital frequency ωD and leads to
warping of digital frequency response.

WARPING
FREQUENCY
From (2), the relationship
between ωA and ωD in fig.2
is non-linear that causing by
the tan function and the
sampling period T. It can be
seen that for small values of
frequency, the relationship
between ωA and ωD is
slightly linear and for larger
values is highly non-linear.

The bilinear z-transform form the s-domain to z-domain is defined
by s 

2 1  z 1
T 1  z 1

(1)

Where T is sampling period, z  e sT and s  j

PRE-WARPING FREQUENCY
•

When converting an analog filter
to a digital filter using bilinear ztransform method will give both
filters have the same behaviour,
but the behaviour is not matched
at all the frequency in s-domain
and digital domain as analysis
above. One way to overcome the
warping frequency is called prewarping frequency.

CONVERTING AN ANALOG LOW PASS FILTER TO
ANOTHER DIGITAL FILTER USING BILINEAR ZTRANSFORM WITH PRE-WARPING
• the block diagram in fig.3, to design a digital filter, first transform
a designed analog low pass filter to an analog filter which the
same class of the digital filter using the frequency
transformations and then apply the equation (4) bilinear ztransform with pre-warping.

1

1
H ((scU1 L ) s 1 1
)
2  
s
H(( 1c z L1  z) 1  t L 1  z1 )
H (t U 1
H((c 2 ) 1 ) U )1
)
ccc  L z
(s  z 1 
1U
1sU UL LU sz   1 tcU t L 1  z 1
 t 1 )
1 z
L
1  cU t L 1  z 1 1  cU t L 1  z 1

The table.1 below, illustrated converting an analog low pass filter to a low pass, high pass,
band pass and band stop digital filter with the bilinear z-transform.

LP to LP

LP to HP

LP to BP

LP to BS

sdomain
H ( s)
H ( s)
H ( s)
H ( s)

Frequency
transformations

H(

H(

s

c
c
s

)

H (c

1  z 1
)
1  z 1

(5)

)

1  z 1
H (t
)
1  z 1

(6)

s 2  LU
)
(U  L ) s
(   L ) s
H( 2U
)
s   LU
H(

z-domain

cU
1  z 1
tL
1  z 1

) (7)
1  cU t L 1  z 1 1  cU t L 1  z 1
1
H(
) (8)
1
cU
1 z
tL
1  z 1

1  cU t L 1  z 1 1  cU t L 1  z 1
H(

BILINEAR Z-TRANSFORM WITH PASCAL’S
TRIANGLE
• This section introduces the bilinear z- transform with pre-waring
to convert an analog transfer function H(s) to a digital transfer
function H(z) involved the Pascal’s triangle. With the Pascal’s
triangle, the matrix equation of the relationship between the
digital coefficients and the analog coefficients can be found easy
to compute and hand-calculated.

THE PASCAL’S
TRIANGLE
One of the most useful applications of the Pascal’s triangle is to find
coefficients and expand the binomial expression
and it can be shown
in fig.4 below.

- The negative Pascal’s triangle:

The positive Pascal’s triangle:



 The coefficients : n

i i 0  n
n

The exp ansion :
 n c n i t i
i


i 0



(9)
(10)



 The coefficients : 1 i n
  i

i 0  n
n

i n
The exp ansion :
  1  i c n  i t i


i 0



(11)
(12)

MATRIX EQUATIONS



[a]  [ P ][  ( A, c , t , m )]
[b]  [P][ ( B, c, t , m )]

The main key is addressed in the next sections considering the matrix
equations of conversion from a low pass to low pass, high pass, band

pass and band stop involved the Pascal’s triangle.

CONVERT AN ANALOG LOW PASS FILTER TO A
DIGITAL LOW PASS FILTER
• The transfer function of an nth-order low pass filter in s-domain
H(s) and the transfer function of a low pass filter in z-domain can
be described as follow:

• From equation (5), the matrix equations of the relationship
between ai , Ai and bi , Bi can be found

All the numbers in the matrix [PLP] can be calculated from the
positive and negative Pascal’s triangle and it can be written as:

Example 1:
Convert a fourth-order Butterworth analog low pass filter has the transfer function
the cut-off frequency of 200hz to digital low pass filter with fs=1kHz.

CONVERT AN ANALOG LOW PASS FILTER
TO A DIGITAL HIGH PASS FILTER
•

The transfer function H(z) of the
digital high pass filter can be
written as the equation (14) and
applying the equation (6), the
matrix equation can described as
equation (16) blow. The matrix
[PHP] is different with the matrix
[PLP] by swapping the first
column to the last column and
the way to find [PHP]i,j.

Example 2:
Convert a fourth-order Butterworth analog low pass filter has the transfer function
the cut-off frequency of 200hz to digital high pass filter with fs=1kHz.

H ( s) 

1
1 + 2.6131s + 3.4142s2 + 2.6131s3 + s4

Converting a low pass to low pass and low pass to high pass
were studied. The first and last column in the matrix [PLP]
and [PHP] are the nth row in the positive and negative Pascal’s
triangle. In the next sections, it is more interesting with
Pascal’s triangle to find the digital coefficients of converting
low pass to band pass and band stop.

DIGITAL BAND PASS FILTER
• The order of transfer function H(z) of the band pass filter is 2n, it
causes from the multiply of number 2 in s2 with the nth-order
when applying the frequency transformation from an nth-order
low pass to a band pass filter. The transfer function H(z) and the
matrix equation can be written as shown below:

• The matrix [PBP] is the same with the matrix [PHP] but it has the
size of (2n+1 ; 2n+1).

If taking out the analog coefficient Ai in each column, the matrix [ΔBP] is the Pascal’s triangle

expansion of (U+L)n. And from that, a formula can be derived for the matrix [ΔBP] as below:

Example 3:

Convert a fourth-order Butterworth analog low pass filter
1
has the transfer function H ( s ) 
2
3
4
1 + 2.6131s + 3.4142s + 2.6131s + s

to digital band pass filter with the lower frequency fL=1khz,
the upper frequency fU=3khz and the sampling frequency fs
=10kHz.

DIGITAL BAND STOP FILTER
•

The matrix equation for
converting an analog low pass to
digital band stop can be derived
from equation (8).

•

The matrix [PBS] is the same the
matrix [PBP]. The matrix [ΔBS] is
similar with the matrix [ΔBP], the
only different is the analog
coefficients Ai , Bi is replaced by
An-i and Bn-i .

Example 4:

Convert a fourth-order Butterworth analog low pass filter
1
has the transfer function H ( s ) 
2
3
4
1 + 2.6131s + 3.4142s + 2.6131s + s

to digital band stop filter with the lower frequency f L=1khz,
the upper frequency fU=3khz and the sampling frequency fs
=10kHz.

GENERAL
FORMULA FOR
CONVERTING AN
ANALOG LOW PASS
TO OTHER DIGITAL
FILTERS
In the fig.4, it can see that
the Pascal’s triangle
involved into the matrix
[ΔBP] and is clearly by
redraw it by fig,5. The Δi is
the sum of all the elements
in the column of Pascal’s
triangle multiply with the
analog coefficient at the
same row.
,

n

Bp

[PN1] 0

 A
0

( U + L )n

Bs
An

1
U

1

A1

2

A2

A3

3

A3

A2

U2

U3

L
L2

2UL

3U2L

3UL2

L3

A1
n

An

A0

Un

Ln
Δ2n-

LptoBp , LptoBs

Δ0

N=2n

ΔM-1

ΔM

ΔM+1

Δ2n

1

Δ0
LptoHp

Δ1

N=n

Δ1

U=0

Δn-1

ΔM=n

LptoLp

Δ0

Δ1

N=n , L = 0

Δn-1 ΔM=n

Fig.5 General formula of the matrix [Δ]

From the fig.5 above, if let L=0, all the elements in Pascal’s triangle have L will equal to
zero, and the [Δ] becomes the matrix [ΔLP] that is the left side edge of the Pascal’s triangle.
And if U=0, the [Δ] = [ΔHP] is the right side edge of the Pascal’s triangle. For the [ΔBS], the
analog coefficients Ai and Bi changes to An-i and Bn-i. So the matrix [ΔBP] can be used for
all converting from Lp to Lp , Lp to Hp , Lp to Bp and Lp to Bs and it can be rewritten as

CONVERT A DIGITAL FILTER TO AN ANALOG
LOW PASS FILTER
Considering the matrix equation of converting an analog low pass filters to another digital filters

Multiply [P] for both sides

Inserting zeros into the Pascal’s triangle as in the fig.6 below is another way to find the matrix [Δ]
shown

Sub (19) into (18),

The equation (20) is the formula to convert the coefficients digital filter to the coefficients
analog filter.

CONCLUSION
In this work, the bilinear z-transform is used to convert an transfer function H(s)
of an analog low pass to the transfer function H(z) of a digital filter and vice versa.
The involving of the Pascal’s triangle made both the direct and inverse
transformation easier for computing and hand-calculated. And all the formulas are

derived and demonstrated. It would be the powerful tools in using all these
formulas for converting analog to digital and from digital to analog filter.

Bilinear z transformaion

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