Analog to Digital Conversion

Transcription

1 Analog to Digital Conversion Lecture No. 7 Dr. Aoife Moloney School of Electronics and Communications Dublin Institute of Technology

2 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 Slide: 1

3 Sampling Quantisation PCM encoding February 2005 Slide: 2

4 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 Quantisation PCM encoder February 2005 Slide: 3

5 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 February 2005 Slide: 4

6 Sampling Theorem: A signal having no spectral components above f m Hz can be determined uniquely by values sampled at uniform intervals of T s s (f s sampling rate), where: 1 T s 2f m or f s 2f m Note: This equation is known as the Nyquist sampling criterion and f s = 2f m 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 Slide: 5

7 X δ (t), where: X δ (t) = n= δ (t nt s ) where, T s is the sampling period of the signal The sampled signal can be expressed as: X s (t) = X(t).X δ (t) = X (nt s ) δ (t nt s ) n= February 2005 Slide: 6

8 This is illustrated in the diagram below. In the diagram the weight (area) of each impulse, X(nT s ), is indicated by the height of the impulse The sampled signal can be examined in the frequency February 2005 Slide: 7

9 domain using the Fourier transform: X s (f) = FT [X(t).X δ (t)] = X(f) X δ (f) = X(f) f s δ (f nf s ) n= = f s X (f nf s ) n= Note: The Fourier transform of an impulse response February 2005 Slide: 8

10 train is well known and is given by: δ (t kt s ) FT f s δ (f nf s ) k= 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 f s Hz February 2005 Slide: 9

11 X(t) W(f) t -B B T s f (a) Waveform and its spectrum X s (t) Low pass filter T s X s (t) t -2f s -f s f s 2f s f B (b) Sampled waveform and its spectrum February 2005 Slide: 10

12 Original Signal: As illustrated in the diagram, if f s 2f m 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 f s < 2f m 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 Slide: 11

13 To avoid aliasing anti aliasing filters can be introduced pre or post sampling Pre sampling filters will have cut off frequencies of f s Post sampling filters will have a cut off frequency of f s f m February 2005 Slide: 12

14 W(f) -B B f (a) Spectrum of unsampled waveform Low pass filter T s X s (t) -2f s -f s f s 2f s B f (b) Spectrum of sampled waveform, fs < 2fm (2B) February 2005 Slide: 13

15 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 f s (1/T s ). February 2005 Slide: 14

16 T s Quantised signal Analogue signal q +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 Slide: 15

17 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 (SN q R). Quantisation Noise: Denoting the quantisation error (i.e. difference between analogue and quantised signals) as e, then assuming linear quantisation (as shown above) February 2005 Slide: 16

18 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: e 2 = q/2 q/2 e 2 p(e)de i.e. e 2 = q2 12 February 2005 Slide: 17

19 Signal to Quantisation Noise Ratio (SN q 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 ( ) Vp 2 Vpp Lq L 2 q 2 = = = The peak signal power to average quantisation noise power, February 2005 Slide: 18

20 SN q R is therefore: SN q R = ( L 2 q 2 4 ( q 2 12 ) ) = 3L February 2005 Slide: 19

21 Conclusion This lecture has looked at the following: Analog to Digital converter (ADC) Sampling Sampling theorem Linear quantisation Signal to quantisation noise ratio (SN q R) February 2005 Slide: 20