Analog to Digital Conversion Lecture No. 7 Dr. Aoife Moloney

School of Electronics and Communications Dublin Institute of Technology

Lecture No. 7: Analog to Digital Conversion

Overview • Now that we have finished the maths and theory we will have a look at baseband transmission. The following topics will be covered: – Analog to digital conversion – Line coding – Detection of baseband signals in noise – Intersymbol interference (ISI) • The next 2 lectures will look at analog to digital conversion and the following: February 2005

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Lecture No. 7: Analog to Digital Conversion

– Sampling – Quantisation – PCM encoding

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Lecture No. 7: Analog to Digital Conversion

Analog to Digital Converter • Remember the analog to digital converter (ADC) that we met in the hypothetical transceiver in Lecture 1 (Communications Engineering handout) • ADCs will generally consist of a sampling circuit, a quantiser and a pulse code modulator ADC Sampling PCM encoder Quantisation

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Lecture No. 7: Analog to Digital Conversion

Sampling • Sampling transforms a continuous waveform into a sequence of samples, with amplitudes derived from the input waveform. This form of sampling is known as PAM (pulse amplitude modulation) and is illustrated in the diagram below. PAM

t

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Lecture No. 7: Analog to Digital Conversion

• Sampling Theorem: A signal having no spectral components above fm Hz can be determined uniquely by values sampled at uniform intervals of Ts s (fs sampling rate), where: 1 ≥ 2fm Ts

or

fs ≥ 2fm

Note: This equation is known as the Nyquist sampling criterion and fs = 2fm is called the Nyquist rate • Sampling can be represented as the product of the signal to be sampled, X(t), with a unit–weight impulse train, February 2005

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Lecture No. 7: Analog to Digital Conversion

Xδ (t), where: Xδ (t) =

∞ X

δ (t − nTs )

n=−∞

where, Ts is the sampling period of the signal The sampled signal can be expressed as: Xs (t) = X(t).Xδ (t) =

∞ X

X (nTs ) δ (t − nTs )

n=−∞ February 2005

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Lecture No. 7: Analog to Digital Conversion

This is illustrated in the diagram below. In the diagram the weight (area) of each impulse, X(nTs ), is indicated by the height of the impulse The sampled signal can be examined in the frequency

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Lecture No. 7: Analog to Digital Conversion

domain using the Fourier transform: Xs (f ) = F T [X(t).Xδ (t)] = X(f ) ⊗ Xδ (f ) = X(f ) ⊗ fs

∞ X

δ (f − nfs )

n=−∞

= fs

∞ X

X (f − nfs )

n=−∞

Note: The Fourier transform of an impulse response

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Lecture No. 7: Analog to Digital Conversion

train is well known and is given by: ∞ X

k=−∞

δ (t − kTs )

FT

←→

fs

∞ X

δ (f − nfs )

n=−∞

The diagram shows the unsampled signal and its spectrum ((a)) and the sampled signal and its spectrum ((b)). Note: The spectrum of the sampled signal is identical to the spectrum of the unsampled signal, but repeated every fs Hz

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Lecture No. 7: Analog to Digital Conversion

W(f)

X(t)

t

-B

f

B

Ts (a) Waveform and its spectrum Low pass filter

TsXs(t)

Xs(t)

t

-2fs

-fs

fs

2fs f

B (b) Sampled waveform and its spectrum

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Lecture No. 7: Analog to Digital Conversion

• Original Signal: As illustrated in the diagram, if fs ≥ 2fm the replicated spectra do not overlap. Thus the original unsampled signal can be regenerated by filtering the sampled signal with a low pass filter selecting the baseband spectrum, as shown in the diagram. • Aliasing: If fs < 2fm the waveform will be undersampled and the replicated spectra of the sampled signal will overlap. The spectral overlap is called aliasing. The receovered signal will be distorted due to the aliasing. As shown in the diagram. February 2005

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Lecture No. 7: Analog to Digital Conversion

To avoid aliasing anti–aliasing filters can be introduced pre or post sampling – Pre–sampling filters will have cut–off frequencies of fs – Post–sampling filters will have a cut–off frequency of fs − fm

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Lecture No. 7: Analog to Digital Conversion

W(f)

-B

f

B

(a) Spectrum of unsampled waveform

Low pass filter

TsXs(t)

-2fs -fs

fs

2fs

f

B (b) Spectrum of sampled waveform, fs < 2fm (2B)

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Lecture No. 7: Analog to Digital Conversion

Quantisation • After sampling, the sampled signal is quantised. Each pulse in the sampled signal is adjusted in amplitude to coincide with the nearest of a finite set of allowed amplitudes. The figure below shows an analogue signal and its corresponding quantised signal, where the signals have been sampled at a sampling rate fs (1/Ts ).

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Lecture No. 7: Analog to Digital Conversion

Ts q Analogue signal Quantised signal

+7q/2 +5q/2 +3q/2 +q/2 0 -q/2 -3q/2 -5q/2 -7q/2

• The step q between quantisation intervals is called the quantile interval • When the quantisation levels are uniformly distributed over the full range of values taken by the sampled signal, the quantiser is called uniform or linear February 2005

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Lecture No. 7: Analog to Digital Conversion

• The difference between the analogue and quantised signals is random and can be thought of as a noise, known as quantisation noise. The ratio of the power of this noise to the peak signal power is known as the signal to quantisation noise ratio (SNq R). • Quantisation Noise: Denoting the quantisation error (i.e. difference between analogue and quantised signals) as e, then assuming linear quantisation (as shown above)

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Lecture No. 7: Analog to Digital Conversion

it follows that the pdf of e, p(e), is uniform and given by:   1/q, for −q/2 ≤ e ≤ q/2; p(e) =  0, elsewhere;

The mean–square quantisation error or noise is therefore: e2 =

Zq/2

e2 p(e)de

−q/2

i.e.

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2 q e2 = 12 Slide:

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Lecture No. 7: Analog to Digital Conversion

• Signal to Quantisation Noise Ratio (SNq R): If L is the number of quantisation levels the peak analogue (i.e. unquantised) signal level is Lq/2. The peak power of the signal (normalised to 1 Ω) is therefore: 2  2  2 2   Lq Lq Vpp 2 Vp = = = 2 2 4 The peak signal power to average quantisation noise power,

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Lecture No. 7: Analog to Digital Conversion

SNq R is therefore: 

L2 q 2 4



SNq R =  2  = 3L

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q 12

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Lecture No. 7: Analog to Digital Conversion

Conclusion This lecture has looked at the following: • Analog to Digital converter (ADC) • Sampling • Sampling theorem • Linear quantisation • Signal to quantisation noise ratio (SNq R)

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