14

D. Rocchesso: Sound Processing

dc component

trapezoid rule

a correspondence between the imaginary axis of the s plane and the unit cir-

cumference of the z plane. A general formulation of the bilinear transformation

is

s = h

1 - z

-1

1 + z

-1

.

(36)

It is clear from (36) that the dc component j0 of the continuous-time system

corresponds to the dc component 1 + j0 of the discrete-time system, and the

infinity of the imaginary axis of the s plane corresponds to the point -1 + j0,

which represents the Nyquist frequency in the z plane. The parameter h allows

to impose the correspondence in a third point of the imaginary axis of the s

plane, thus controlling the compression of the axis itself when it gets transformed

into the unit circumference.
A particular choice of the parameter h derives from the numerical integration

of differential equations by the trapezoid rule. To understand this point, consider

the transfer function (32) and its relative differential equation that couples the

input variable x
s

to the output variable y

s

dy

s

(t)

dt

- s

a

y

s

(t) = ax

s

(t) .

(37)

If we sample the output variable with period T we can write

y

s

(nT ) = y

s

(nT - T ) +

nT

nT -T

y

s

( )d ,

(38)

where y

s

=

dy

s

(t)

dt

, and integrate the (38) with the trapezoid rule, thus obtaining
y

s

(nT ) y

s

((n - 1)T ) + ( y

s

(nT ) + y

s

((n - 1)T )) T /2 .

(39)

By replacing (37) into (39) and setting y(n) = y
s

(nT ) we get a difference equa-

tion represented, in virtue of the shift theorem 1.2, by the transfer function
H(z) =

a(1 + z

-1

)T /2

1 - s

a

T /2 - (1 + s

a

T /2)z

-1

,

(40)

which can be obviously obtained from H

s

(s) by bilinear transformation with

h = 2/T .

It is easy to check that, with h =

2

T

, the continuos-time frequency f =

1

T

maps into the discrete-time frequency =

2

, i.e. half the Nyquist limit. More

generally, half the Nyquist frequency of the discrete-time system corresponds to

the frequency f =

h

2

of the continuous-time system. The more h is high, the

more the low frequencies are compressed by the transformation.

To give a practical example, using the sampling frequency F

s

= 44100Hz and

h =

2

T

= 88200, the frequency that is mapped into half the Nyquist rate of the

discrete-time system (i.e., 11025Hz), is f = 14037.5Hz. The same transforma-

tion, with h = 100000 maps the frequency f = 15915.5Hz to half the Nyquist

rate. If we are interested in preserving the magnitude and phase response at

f = 11025Hz we need to use h = 69272.12.

1.6

Quantization

With the adjectives "numeric" and "digital" we connote systems working on

signals that are represented by numbers coded according to the conventions of