Digital Telephony1

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

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

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)

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

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

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

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

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 

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)

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=  

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

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)

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

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

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

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 

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

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

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 =   

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

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)

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 log 10 (X m /  x ) Quantizer + x(n) x q (n) x(n) x q (n) e q (n)

Digital Telephony 24 Uniformly Encoded PCM X/Xm Number of bits per sample dB Signal to Quantiiation Noise Ratio (dB)

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.

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

Digital Telephony 27 Segment Approximation Input Sample Values Uniform quantization

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

Digital Telephony 29 Performance of a  Encoder dB Signal Power of sinewave (dBm0) Signal-to-quantization noise ratio (dB) 8 bit  bit  100 Piecewise linear 8 bit 

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

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

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

Digital Telephony 33 Plesiochronous Transmission Rates 64 kbits/s Japanese StandardNorth America Standard European Standard 1544 kbits/s 2048 kbits/s 8448 kbits/s kbits/s kbits/s kbits/s 6312 kbits/s kbits/s kbits/s kbits/s kbits/s x24 x30 x4 x3 x4 x3 x4 x5x7 x6 x4 x3

Digital Telephony 34 Plesiochronous Digital Hierarchy

Digital Telephony 35

Digital Telephony 36 Plesiochronous Digital Hierarchy F The output of the M12 multiplexer is operating 136 kbs faster than the agragate rate of four DS 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

Digital Telephony 37 Makeup of a DS2 Frame M C F C C F M C F C C F 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

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

Digital Telephony 39 Digital Transmission Systems

Digital Telephony 40 PCM System Enhancements F North America  Superframe of 12 DS0’s has a sync sequence for odd ( 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