5. Continuous & Discrete Systems

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5. CONTINUOUS & DISCRETE SYSTEMS • Discrete systems (digitally-implemented systems) have many parallels with continuous systems. • Basic notation: INPUT x (t)

ANALOGUE SYSTEM

OUTPUT y (t)

h (t) INPUT x (nT) or x [n]

DISCRETE SYSTEM

OUTPUT y (nT) or y [n]

h (nT) or h [n]

• Basic elements: x1 [n]

⊕

(i) Summer

x1 [n] + x2 [n]

x2 [n] (ii) Multiplier

x [n]

(iii) Delay

x [n]

a

a x [n]

z-1 1.1

x [n-1]

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Consider now Sampling & Quantisation and their Effects 5.1 Sampling Essential 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: x * (t ) = x(t ) f s (t ) = x(t )

∞

∑δ (t − mT )

m = −∞

input

infinite series of impulsive sampling pulses 1.2

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(i) Sampling and the δ-Function Terminology: δ (t − τ ) = impulse at t = τ

δ ( t ) = impulse at time origin

τ The strength (area) of a unit impulse response is unity. The product of a waveform x(t) and a unit impulse δ(t) can be viewed as a “masking” process: and

∫

∞

−∞

δ (t )x(t )dt =x(0)

τ+

∫τ

δ ( t − τ ) dt = 1 and −

since:

∫

∞

−∞

δ (t − τ )x(t )dt =x(τ )

δ (t − τ ) = 0

Also, from basic Fourier Transform theory: and

δ (t ) ⇔ 1

Aδ ( t ) ⇔ A

1.3

for all t ≠ τ

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(ii) Fourier Series for Sampling Signal fs(t) is periodic with period T and can therefore be represented by a Fourier Series: ∞

∑ δ ( t − mT )

f s (t ) =

=−∞ m is an integer. Using complex exponential mform:

f s (t ) =

∞

∑

n =−∞

n is an integer, where:

1 T

e jnω s t

ω s = 2π T

1 ∴x * (t ) = T

Using standard Fourier result: where:

∞

∑ x(t )e

n = −∞

(shift)

f ( t ) e bt ⇔ F ( jω − b )

f ( t ) ⇔ F ( jω ) Taking Fourier Transform of both sides of (6) gives:

where:

jnω s t

1 X * ( jω ) = T x ( t ) ⇔ X ( jω ) 1.4

∞

∑ X[ j ( ω − nω )] s

n =−∞

( 6)

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(iii) Spectrum of Sampled Signal 1 Recall: X * ( jω ) = T

∞

∑ X[ j ( ω − nω )] s

n =−∞

x ( t ) ⇔ X ( jω )

If X(jω) has the form:

The spectrum of x*(t) is:

i.e. X*(jω) is periodic with period ωs in the frequency domain. 1.5

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(iii) Spectrum of Sampled Signal (contd.) From Fourier Theory:  multiplication  of waveforms in  the time domain

  convolution of  ⇔  spectra in the   frequency domain

  

Sampling operation in the time domain described by:

x *( t ) = f s ( t ) x ( t ) and in the frequency domain by:

X *( jω ) = Fs ( jω ) ⊗ X ( jω ) i.e. convolution of sampling signal and input spectra.

1.6

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5.2 Sampling (Nyquist) Theorem Sampled signal spectrum:

(a) If ωs > 2ωc , then spectral components do not overlap and X(jω) can be recovered by low-pass filtering. (b) At ωs = 2ωc , LPF needs to be an ideal “brick wall” filter. (c) If ωs < 2ωc , spectral components overlap and “aliasing” occurs:

Signal cannot then be recovered unambiguously from its samples and sampling becomes irreversible. 1.7

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5.2 Sampling (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:

ω s ≥ 2 (ω H − ω L )

where W is the signal “bandwidth”.

ω s ≥ 2W

In practice, sampling rates must be above the minimum to allow for: (a) non-brick wall spectra and filters; (b) non-ideal sampling pulses. 1.8

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5.3 Quantisation This is also an essential aspect of the process of analogue-to-digital conversion. Approximates a continuous signal x(t) with a discrete-level signal xQ(t), e.g.

+ ∆V − ∆V

It is seen that:

2

2

− ∆V ∆V ≤ e( t ) ≤ 2 2

∆V  ∆V     mi −  ≤ x ( t ) ≤  mi +    2  2 

The quantiser output level is mi when: 1.9

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

e = 2 i

∫

mi + ∆V 2

mi − ∆V 2

( x − mi ) 2 p( x ) dx

where p(x) is the amplitude PDF of x(t). If ∆V << [ Total amplitude range of x(t) ] , then: p(x) ≈ p(mi )

p( mi )

(mi

− ∆2V )

mi

(mi

+ ∆2V )

i.e. PDF remains approximately constant over interval ∆V centred on mi. Hence, mean square error for i th level:

e = p( mi ) ∫ 2 i

mi + ∆V 2

mi − ∆V 2

( x − mi )2 dx

1.10

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5.3 Quantisation (contd.) ei2 = p( mi ) ∫

mi + ∆V 2

mi

− ∆V

2

( x − mi )2 dx

Let (x - mi ) = y ; thus dx = dy and limits of integration become: ∆V

∆V

ei2 = p( mi ) ∫− ∆V y 2 dy 2

=

2

( ∆V )3 = p( mi ) 12 Total mean square error for all levels:

 y3  2 p( mi )    3  − ∆V 2

1 ( ∆V ) 3 p( mi ) ∑ 12 i 1 = ( ∆V ) 2 ∑ [ ∆V p( mi ) ] 12 i

e2 =

[ ∆V p(mi) ] is approximate area of strip of width ∆V centred on mi . ∴

∑ [ ∆V p(m ) ] i

i

Hence:

≈ total area under PDF curve =1

( ∆V ) 2 e = 12 2

for a “linear” (equi-interval) quantiser. 1.11

±

∆V 2

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5.4 Quantisation 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, if x(t) has peak amplitude ±A, then: 1 A= [ q (∆V ) ] 2

The ratio of:

 Peak input signal level   RMS quantisation noise   

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:

2

q   1  ≈  (∆V )    2   2 2

2

2 1 q  (∆V ) γ =  (∆V )  ÷ 2 2 12  3 = q2 2

1.12

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5.4 Quantisation SNRs (contd.) (b) Uniform Amplitude Distribution (e.g. ramp or triangular waveform) Mean power of quantiser output, taken over complete output range of q levels, is the mean of the individual level powers, i.e.

1 = q

Total mean power

q

∑ i =1

 i  ( ∆ V )  2 

2

( ∆V ) 2 q 2 = ∑i 4q i=1 This is a standard series summation with the value: q

∑ i2 = i =1

q ( q + 1) ( 2q + 1) 6

q3 i ≈ ∑ 3 i =1 q

If q >> 1 ∴ Mean Power Hence:

2

( ∆V ) 2 q 3 = 4q 3 ( ∆V ) 2 2 ( ∆V ) 2 γ = q ÷ 12 12 = q2 1.13

=

( ∆V ) 2 2 q 12

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5.4 Quantisation SNRs (contd.) (c) Rectangular Input (Equal mark:space) Two levels at extremes of quantiser range: Hence:

( ∆V ) 2 q2 2 γ = ( ∆V ) ÷ 4 12

q ±  ( ∆V 2 =

 ) 

3q2

(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

γ = =

1 q ( ∆V ) 4 2

1 2 q ( ∆V ) 2 64 ( ∆V ) 2 1 2 2 q ( ∆V ) ÷ 64 12

3 2 q 16

γ can therefore be calculated for different input types and different values of q.

1.14