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5. CONTINUOUS & DISCRETE SYSTEMS Discrete systems (digitally-implemented systems) have many parallels with

continuous systems. Basic notation:

Basic elements:

x1 [n]

(i) Summer x1 [n] + x2 [n]

x2 [n]

(ii) Multiplier x [n] a x [n]

(iii) Delay x [n] x [n-1]

ANALOGUE SYSTEM

DISCRETE SYSTEM

h (t)

h (nT) or h [n]

INPUT

OUTPUT

OUTPUT

INPUT

x (t)

x (nT) or x [n]

y (t)

y (nT) or y [n]

z-1

a

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Consider now Sampling & Quantisation and their Effects

5.1 SamplingEssential in the conversion of analogue signals to digital form and vice-versa.

Consider the following arrangement:

Sampler can be represented as a multiplier. Thus, sampled signal is given by:

input infinite series ofimpulsive sampling pulses

=

=m

mTttx )()( )()()(* tftxtx s=

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(i) Sampling and the -FunctionTerminology:

The strength (area) of a unit impulse response is unity.

The product of a waveformx(t) and a unit impulse (t) can be viewed as a maskingprocess:

and

since: and

Also, from basic Fourier Transform theory:

and

= )0()()( xdttxt

= )()()( xdttxt

( ) ( )t dt t t = =

+

1 0 for all

( ) ( )t A t A 1

( )t = impulse attime origin

( )t

t

=

=impulse at

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(ii) Fourier Series for Sampling Signal

fs(t) is periodic with period Tand can therefore be represented by a Fourier Series:

m is an integer. Using complex exponential form:

n is an integer, where:

Using standard Fourier result: (shift)

where:

Taking Fourier Transform of both sides of (6) gives:

where:

f t t mT sm

( ) ( )= =

f ts Tn t

n

s( ) ==

1 ej

sT

= 2

)6(e)(1

)(*j

=

=n

tn stxT

tx

f t F bbt( ) ( )e j

f t F ( ) ( ) j

[ ]XT

X n sn

* ( ) ( )j j = =

1

x t X ( ) ( ) j

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(iii) Spectrum of Sampled Signal

Recall:

IfX(j) has the form:

The spectrum ofx*(t) is:

i.e.X*(j) is periodic with period s in the frequency domain.

[ ]XT

X n

x t X

s

n

* ( ) ( )

( ) ( )

j j

j

=

=

1

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(iii) Spectrum of Sampled Signal (contd.)

From Fourier Theory:

Sampling operation in the time domain described by:

and in the frequency domain by:

i.e. convolution of sampling signal and input spectra.

multiplicationof waveforms in

the time domain

convolution ofspectra in the

frequency domain

x t f t x t s*( ) ( ) ( )=

X F X s*( ) ( ) ( )j j j =

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5.2Sampling (Nyquist) TheoremSampled signal spectrum:

(a) If s > 2c , then spectral components do not overlap andX(j) can be recovered by

low-pass filtering.(b) At s = 2c , LPF needs to be an ideal brick wall filter.(c) If s < 2c , spectral components overlap and aliasing occurs:

Signal cannot then be recovered unambiguously from its samples and sampling

becomes irreversible.

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5.2Sampling (Nyquist) Theorem (contd.)In its simplest form, the sampling theorem states that a waveform should be sampled at a

rate which is at least twice its highest significant frequency component if it is to be

recoverable from the samples. This applies to low-pass situations, i.e.

A more general result applies to band-pass signals, i.e.

Here: or: where Wis the signal bandwidth.

In practice, sampling rates must be above the minimum to allow for:

(a) non-brick wall spectra and filters;

(b) non-ideal sampling pulses.

s H L 2 ( ) s W 2

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5.3QuantisationThis is also an essential aspect of the process of analogue-to-digital conversion.

Approximates a continuous signalx(t) with a discrete-level signalxQ(t), e.g.

It is seen that:

The quantiser output level is miwhen:

Ve t

V

2 2( )

mV

x t mV

i i

+

2 2

( )

+V2

V2

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5.3Quantisation (contd.)The mean square error voltage associated with level miis:

wherep(x) is the amplitude PDF ofx(t).

IfV

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5.3Quantisation (contd.)

Let (x - mi) = y; thus dx = dyand limits of integration become:

Total mean square error for all levels:

[ Vp(mi) ] is approximate area of strip of width Vcentred on mi.

Hence:

for a linear (equi-interval) quantiser.

e p m x m dxi i im

m

iV

iV

2 2

2

2=

+

( ) ( )

V

2

e p m y dy p my

p mV

i i i

V

V

i

V

V

2 23

2

2

3

2

2

3

12

= =

=

( ) ( )

( )( )

[ ]

e V p m

V V p m

i

i

i

i

2 3

2

1

12

1

12

=

=

( ) ( )

( ) ( )

[ ] =

V p mii

( ) total area under PDF curve

1

eV2

2

12=

( )

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5.4Quantisation Signal-to-Noise Ratios (SNRs)Assume that the quantiser has q levels, and that peaks of input signal always match the

complete quantiser input range. Hence, ifx(t) has peak amplitude A, then:

The ratio of:

is taken as a measure of quantisation amplitude SNR. The power SNR will be the square

of this.

Consider different input types:

(a) Sinusoid

For sinusoidal input, output will also be approximately sinusoidal if q>>1.

Mean square output:

Hence power SNR:

[ ])(21

VqA =

Peak input signal level

RMS quantisation noise

22

2

1)(

2

V

q

2

22

2

3

12

)()(

22

1

q

VV

q

=

=

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5.4Quantisation SNRs (contd.)(b) Uniform Amplitude Distribution (e.g. ramp or triangular waveform)

Mean power of quantiser output, taken over complete output range ofq levels, is the

mean of the individual level powers, i.e.

Total mean power

This is a standard series summation with the value:

Ifq >> 1

Mean Power

Hence:

=

=

=

=

1

2

4

2

1

22

1

q

iV

V

qi

i

q

i

q

( )

( )

iq q q

i

q2

1

1 2 1

6= = + +( ) ( )

iq

i

q2

1

3

3=

= =( ) ( ) V

q

q Vq

2 3 2

2

4 3 12

=

=

( ) ( ) Vq

V

q

2

2

2

212 12

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5.4Quantisation SNRs (contd.)

(c) Rectangular Input (Equal mark:space)

Two levels at extremes of quantiser range:

Hence:

(d) Gaussian Input (e.g. speech)

Approximation: Peak 4 x (RMS Value) Limit quantiser input amplitude to: 4 x (RMS Value). Thus:

RMS

Mean power

and

can therefore be calculated for different input types and different values ofq.

qV

2( )

= =q

VV

q2

2

2

2

4 123( )

( )

= 14 2

q V( )

=1

64

2 2q V( )

=

=

1

64 12

316

2 2

2

2

q VV

q

( )( )