1. Digital Telephony1

2. 2 Analog/digital systems Analog signal -voltage -speech -pressure SP Analog Sampler Discrete signal F s  F max Quantiz- er Error is introduced Digital signal A/D converter or data from -tape -simulations -digital devices DSP Digital Signal Processor -digital computer -dedicated dig. hw -programmable hw Digital signal D/A Analog signal

3. Digital Telephony 3 Issues F Reconstruction accuracy  Conditions for perfect reconstruction F Digital signal is not just an approx. representation of an analog signal  Could be generated digitally  The processing being performed may not be realizable in analog F The theory of discrete time signal processing is independent of continuous

4. Digital Telephony 4 Digital vs. analog processing F DSP implementations are flexible, programmable and modular F More precise and repeatable F Performance and cost effectiveness (riding the microelectronics wave) F Direct mapping of mathematical expressions with less approximation possible (enables sophisticated algorithms)

5. Digital Telephony 5 Digital vs. analog... F Digital hardware can be multiplexed better than analog. Allows integration of multiple operations and services on a h/w platform F Digital storage is more reliable, cheaper and more compact

6. Digital Telephony 6 On the other hand F Analog SP still offers higher bandwidth F Higher dynamic range F Can be very low power

7. Digital Telephony 7 Analog to Digital Conversion F To convert “real-world” analog signals to digital signals for processing F Sampling F Quantizing and coding Xa(t)Xa(t) X [n] Xq[n]Xq[n] Sampler Quantizer and Coder Analog signal Discrete signal Digital signal

8. Digital Telephony 8 Sampling F Uniform  One sample every T seconds (ideally)  x[n] = x a (nT),  n   Sampling period: T  Sampling frequency: F s =1/T  Assume: x a (t) = Acos( 2  Ft+  ) = Acos(  t+  )  Then: x[n] = Acos[2  FnT+  ] = Acos[  Tn+  ] = Acos[  n+  ], where   T is called the normalized or discrete domain frequency

9. Digital Telephony 9 F f = F/ F s must be rational in order for x[n] to be periodic F If f = k/N, then x[n] is periodic with period N  Now, x a (nT) = Acos(  Tn+  ) = Acos((  +2  k/T  Tn+  ) is periodic in  with period 2  /T  Also, x[n] = Acos[  n+  ] = Acos[(  +2  k) n +  ] is periodic in  with period 2 

10. Digital Telephony 10 xa(t)xa(t) n=   S(t) =   (t  nT)  x s (t) =  x a (nT)   (t  nT) n=   convert to discrete sequence x[n] = x a (nT)

11. Digital Telephony 11 F Let us look at the continuous time Fourier transform of x s (t) X s (j  ) = X a (j  ) * S(j  ) S(j  ) =   k  s  X s (j  ) =  X a (j  kj  s ) 2T2T  k=  1 2  1T1T k=  

12. Digital Telephony 12  Thus, X a (j  ) must be bandwidth limited  If the max frequency in X a (j  ) is  N, then the sampling rate  s  2  N  ensures no information is lost due to aliasing F This sampling rate is known as Nyquist rate F A lower sampling rate causes a distortion of the signal due to Aliasing F If no Aliasing occurs, the signal can be perfectly reconstructed by passing through an ideal low pass filter with

13. Digital Telephony 13 Reconstruction X r (j  ) = H r (j  ) X s (j  ) if  N  c  (  s  N ) then X r (j  ) = X c (j  ) Hr(j)Hr(j)  c cc  s >2   Xs(j)Xs(j)

14. Digital Telephony 14 Reconstruction Frequency response of ideal reconstruction filter  c cc T Impulse response of ideal reconstruction filter H r (j  ) = { T,  c  c 0, otherwise h r (t)= sin  t/T  t/T

15. Digital Telephony 15 Reconstruction  X r (j  ) = H r (j  ) X s (j  ) F x r (t) = x s (t) * h r (t) = [  k  x a (kT)  t-kT    h r (t) =  k  x a (kT) h r (t-kT)  k  x a (kT)    sin  t-nT)/T  t-nT)/T

16. Digital Telephony 16 xa(t)xa(t) xs(t)xs(t) hr(t)hr(t) xr(t)xr(t)

17. Digital Telephony 17 Sampling theorem  If the highest frequency contained in a signal x a (t) is  0 and the signal is uniformly sampled at a rate  s  0, then x a (t) can be exactly recovered from its sample values using the interpolation function and then x a (t) =  k  x a (kT) h r (t-kT), where {x a (kT) } are the samples of x a (t), and T=2  s h r (t)= sin  t/T  t/T 

18. Digital Telephony 18 Quantization and coding F Quantization:  Converting discrete time signal to digital  x q (n) =Q [x(n)]  Quantization step

19. Digital Telephony 19 x Q(x)               

20. Digital Telephony 20 Quantization F Rounding: Assign x[n] to the closest quantization level F Quantization error e q [n] = x q [n] - x[n]  e q [n]   Uniformly distributed u mean = 0  variance =   

21. Digital Telephony 21 Quantization F Range of quantizer: x max -x min F Quantization levels: m F Assuming uniform quantization   =  X m / (m-1) where X m = (x max -x min )/2 is called the full-scale level of the A/D converter m-1 x max -x min

22. Digital Telephony 22 Coding F Coding is the process of assigning a unique binary number to each quantization level  Number of bits required  log 2 m F Alternatively, given b+1 bits  x max -x min )/2 b+1 =X m /2 b F For A/D devices, the higher F s and m, the less the error (and the more the cost of the device)

23. Digital Telephony 23 F Assuming dynamic range of A/D converter is larger than signal amplitude  SNR = 10 log 10 (  x  e ) = 10 log 10 (  x    ) = 10 log 10 (12.2 2b  x /  X m  ) =6.02b +10.8 + 20 log 10 (X m /  x ) Quantizer + x(n) x q (n) x(n) x q (n) e q (n) 22 2 2 2

24. Digital Telephony 24 Uniformly Encoded PCM X/Xm 20 40 60 80 -40-30-20-10 0 Number of bits per sample 13 12 11 10 9 8 dB Signal to Quantiiation Noise Ratio (dB)

25. Digital Telephony 25 Example F What is the minimum bit rate that a uniform PCM encoder must provide to encode a high fidelity audio signal with a dynamic range of 40 dB? Assume the fidelity requirements dictate passage of a 20-kHz bandwidth with a minimum signal-to-noise ratio of 50 dB. For simplicity, assume sinusoidal input signals.

26. Digital Telephony 26 Companding F Companded PCM with analog compression and expansion A/D Compression Linear PCM Encoder Input Signal D/A Linear PCM Decoder Expansion Output Signal Compressed Digital Codewords

27. Digital Telephony 27 Segment Approximation Input Sample Values 000 001 010 011 100 101 110 111 Uniform quantization

28. Digital Telephony 28 T1 Channel Bank A/D D/A T1 transmission Line Analog Inputs 1 2 24 Eigth bits per PCM code word companding functions with mu=255

29. Digital Telephony 29 Performance of a  Encoder 10 20 30 40 -70-60-50-40-30 dB -20-1003 Signal Power of sinewave (dBm0) Signal-to-quantization noise ratio (dB) 8 bit  255 7 bit  100 Piecewise linear 8 bit  255 22 27 33

30. Digital Telephony 30 Total Noise Power Signal Power relative to full-load signal (dBm0) -70 15 dB at which persons find communication difficult Signal-to-total noise noise ratio 1010 20 30 40 -60-50-40-30 dB -20-1003 30 dB required for good communication 40 dB range of possible signals

31. Digital Telephony 31 Error Performance F Fewer than 10% of 1 min intervals to have BER worse than 10E-6 F Fewer than 0.2% of 1 sec intervals to have BER worse than 10E-3 F 92% error free sec

32. Digital Telephony 32 DS1 Signal Format  (8x24)+1=193 bits in 125  s F 193 x 8000 = 1.544 Mbs F Bit “robbing” technique used on each sixth frame to provide signaling information

33. Digital Telephony 33 Plesiochronous Transmission Rates 64 kbits/s Japanese StandardNorth America Standard European Standard 1544 kbits/s 2048 kbits/s 8448 kbits/s 34368 kbits/s 139264 kbits/s 564992 kbits/s 6312 kbits/s 44736 kbits/s 274176 kbits/s 32064 kbits/s 97728 kbits/s x24 x30 x4 x3 x4 x3 x4 x5x7 x6 x4 x3

34. Digital Telephony 34 Plesiochronous Digital Hierarchy

35. Digital Telephony 35

36. Digital Telephony 36 Plesiochronous Digital Hierarchy F The output of the M12 multiplexer is operating 136 kbs faster than the agragate rate of four DS1 6.312 vs 4x1.544=6.176 F M12 frame has 1176 bits, i.e. 294-bit subframes ; each subframe is made of up of 49-bits blocks; each block starts with a control bit followed by a 4x12 info bits from four DS1 channels

37. Digital Telephony 37 Makeup of a DS2 Frame M1 01 02 03 04C1 01 02 03 04F0 01 02 03 04C2 01 02 03 04C3 01 02 03 04F1 01 02 03 04 M1 01 02 03 04C1 01 02 03 04F0 01 02 03 04C2 01 02 03 04C3 01 02 03 04F1 01 02 03 04 Bit stuffing F 4 M bits (O11X X=0 alarm) F C=000,111 bit stuffing present/absent F nominal stuffing rate 1796 bps, max 5367

38. Digital Telephony 38 Regenerative Repeaters Amplifier Equalizer Input Timing recovery Regenerator Output F Spacing between adjacent repeaters

39. Digital Telephony 39 Digital Transmission Systems

40. Digital Telephony 40 PCM System Enhancements F North America  Superframe of 12 DS0’s has a sync sequence 101010 for odd (001110 for even frames) F Extended superframe  24 frames - (4 S bits for frame allignment signal); 6 S bits for CRC-6 check; the rest 12 constitute 4 kbs data link