0. Chapter 3: Baseband Demodulation/Detection

1. Idiots, it’s trade-off!! 풍선효과 Bitrate R Bit error PB Bandwidth W Power
Eb/No
Bit error PB
Bandwidth W
Idiots, it’s trade-off!!

2. White Noise  ACS  PSD ( Ex ) Thermal noise : 0 ~ 1012 Hz
 AWGN ( Additive White Gaussian Noise ) Channel
- Infinite Bandwidth :
 Bandlimited AWGN channel
- Pass through a specific channel
- If AWGN is correlated with one of a set of orthonormal functions
the variance of the correlator output is given by
( Why ? Appendix C )

3. Three Major Functions of Digital Receiver
 Filtering : Reduce unwanted noise and interference
- Recover a baseband pulse with the best possible
SNR, free of any ISI (Inter-Symbol Interference
- Matched filter or Correlator
- Equalizing filter for channel – induced ISI
 Sampling : Get the best signal components
 Decision : Reduce the probability of error
( Note ) Detection : Decision – making process

4. Functional Block for Digital Demodulation / Detection Procedure

5. Vectorial Representation of Signal Waveforms (1)
 N-dimensional orthogonal space
Dfn : A space characterized by a set of N linearly independent functions ,
called basis function.
 Orthogonal and orthonormal functions –
, Kronecker delta function –
– If , then are called orthonormal functions.
Cf)

6. Vectorial Representation of Signal Waveforms (2)
 Any arbitrary function in the space can be expressed as a linear combination of
N orthogonal waveforms, such that –
M symbols
– Compact form
– Coefficient form

7. Vectorial Representation of Signal Waveforms (3)
 Observation :
– The set of signal waveforms , can be viewed as a set of vectors
(Ex.)
The task of the receiver is to decide whether has a close “resemblance”
to the prototype
The analysis of all demodulation of detection schemes involves this concept of distance
between a received waveform and a set of possible transmitted waveforms.
Refer to Fig. 3.4

8. Example of Vector Representation

9. Waveform Energy Eq. (3.13) ~ Eq.(3.17)
Energy of the waveform over a symbol duration
Eq. (3.13) ~ Eq.(3.17)
If , then
Average symbol energy :
Average bit energy :

10. Variance of White Noise
▷ white Gaussian noise process, n(t), with zero mean and two-sided power spectral density,
▷ Noise variance is infinite, filtered AWGN is finite
▷ The output of each correlator, , t=T
▷ mean of

11. ▷ variance of
▷ n(t) is zero-mean process
▷ The autocorrelation function
▷ If n(t) is assumed stationary, then Rn(t,s) is only a function of the time difference,

12. ▷ For a stationary random process, the power spectral density, Gn(f), and the autocorrelation function,Rn( ), form a Fourier transform pair.
▷ Since n(t) is white noise,
So,
 Inverse FT 1

13. Signal Representation
 Transmitted signal :
convolution
 Received signal :
 Output of receiving filter :
 Output of sampler at t = T :
: Gaussian random variable ( n0 = n(T) )
: Gaussian random variable ( a1 = a1(T) , a2 = a2(T) )

14. Conditional pdf p ( z/s )
 - If s1(t) is transmitted , then
- If s2(t) is transmitted , then σo o no
 Conditional pdf : -
: Likelihood function of s1
: Likelihood function of s2 - a1 a2
 Fig

15. Eb / N0
 Analog communications : S/N ( or SNR ) figure of merit ( power signal )
 Digital communications : Eb/N figure of merit
- Eb : bit energy , N0 : one – sided PSD S : signal power
 Why Eb/N0 is a natural figure of merit ?
- Focusing on one symbol , the power ( averaged over all time ) goes to zero .
- Hence , power is not a useful way to characterize a digital waveform .
- The symbol energy ( power integrated over Ts ) is a more useful parameter
for characterizing digital waveforms .
- Tb : bit duration[sec] S : signal power[Watt] N : noise power[Watt] 

16. 3.2 Detection of Binary Signal in Gaussian noise
Demodulation and Detection
The received signal over Gaussian channel

18. Decision-making criterionH1 H2 a1 a2
is equal

19. ln 1 = 0
Minimum error criterion

20. BER ▷ Error Probability a1 a2
and because of the symmetry of the probability density function
BER a1 a2

21. Complementary error function
Right half of N(0,1) x

22. Matched Filter ( MF ) ( 1 )
 A linear filter designated to provide the maximum SNR
at its output for a given transmitted symbol waveform .
 AWGN Channel , transmitted signal

23. Matched Filter
Linear Filter designed to provide the maximum SNR at its output

24. s(t) T
h(t) T
Matched filter일 때, S/N이 최대값이 됨을 증명했다.

25. Correlator Realization of the MF
 At t = T , : Cross – correlation of with
( Observation ) Integration – and – Dump Filter
 For NRZ Waveform ,
( Note ) The correlator output and the matched filter output are
the same only at time t = T ( Refer to Fig. 3.7 )

26. Matched Filter ▷ Correlation Realization of the Matched Filter
MF output
When t=T

27. Equivalence of Matched Filter and Correlator

28. Probability of Bit Error ( PB ) for Binary Signaling (1)
Cross-correlation coefficient -1<ρ<1
 ρ =0 if uncorrelated a1 a2
( ) Output SNR at time t = T :

29. Probability of Bit Error ( PB ) for Binary Signaling (2)
for antipodal signaling s2 s1 s2
for orthogonal signaling s1

30. Comparison of Binary Signaling
At the same bit energy Eb and noise condition No,
Compare

31. Unipolar Signaling (1)
 Signal – –

32. Unipolar Signaling (1)
 Detection Process

33. Bipolar Signaling (1) Decision t – for binary 1 – for binary 0 A T 2T

34. Bipolar Signaling (2)  Average bit energy :   Refer to Fig. 3.14
10-2
10-4
3dB -1 14

35. 3.2.5.3 Signaling Described with Basis Functions (1)
Basis functions  orthonormal functions
 Unipolar Signaling – – –

36. 3.2.5.3 Signaling Described with Basis Functions (2)
 
 Detection process

37. 3.2.5.3 Signaling Described with Basis Functions (3)
 Bipolar Signaling – – 
 Detection process

38. Observations on Binary Signaling
 We can see that a 3-dB error performance improvement for bipolar
signaling compared with unipolar signaling .
 Bandpass antipodal signaling ( e.g. , BPSK ) has the same PB
performance as baseband antipodal signaling ( e.g. , bipolar pulses ) s1 s2
with MF detection .
 Bandpass orthogonal signaling ( e.g. , orthogonal FSK ) has the same
PB performance as baseband orthogonal signaling ( e.g. , unipolar pulses ) s1 s2

39. Bandlimited Channel
 So far , digital communication over on AWGN channel
– No bandwidth requirement and/or channel distortion
 Now , digital communication over a bandlimited baseband channel
– Bandwidth constraint and/or channel distortion
– Modeled as a linear filter channel with a limited bandwidth
– Telephone channels , microwave , satellite etc.

40. Linear Filter Channel
 More stringent requirements on the design of modulation signals
 Preclude the use of rectangular pulses at the modulator output
 Distort the transmitted signal
 Cause the intersymbol interference ( ISI ) at the demodulator
Frequency responses of channel are distorted , thus non-flat or
frequency
selective channels .
 Increase PB
Requires channel equalizers to compensate for the distortion
caused by
the transmitter and the channel .

41. Our dilemma!! Infinite bandwidth FT f t FT t f Inter-symbol
Infinite duration
and non-causal FT t f
Inter-symbol
interference
양쪽에서 조금씩 타협

42. Typical Baseband Digital System
System Transfer function :
Transmit Channel Receiving
Equivalent model with two pulses

43. 3.3 Inter-symbol Interference ( ISI )
ISI : Due to the effects of system filtering , the
received pulses can overlap one another.
The tail of a pulse can smear into adjacent symbol
Interfere with the detection process
Degrade the error performance
Even in the absence of noise, the effects of filtering
and channel-induced distortion lead to ISI.

44. Ideal Nyquist Filter for Zero ISI (1)
Theoretical Minimum Nyquist BW = –
: Symbol rate [ symbols / sec ]
Ideal Nyquist pulse :
infinite tail  find practical one

45. Ideal Nyquist Filter ( apulse ) for Zero ISI (2)
Observation :
(i) Even though has long tails, a tail pass through zero
amplitude at t = T when is to be sampled.
(ii) If the sample timing is perfect, thee will be no ISI degradation
introduced.
Nyquist filters are not realizable
since they have the infinite filter length.
Among the class of Nyquist filters, the most popular ones are
the raised cosine and the square root-raised cosine filters.

46. Pulse shape to Reduce ISI
Nyquist filters provide zero ISI only when the sampling
is performed at exactly the correct sampling time. When
the tails are large, small timing errors will result in ISI
One frequently used transfer function belonging
to the Nyquist close ( zero ISI at the sampling points)
is called the raised-cosine filter.

47. Raised – Cosine Filter ( R-C Filter )

48. Raised – Cosine Filter ( R-C Filter ) (2)
Impulse Response :
: Minimum Nyquist Filter
: roll-off factor
: symbol rate
Baseband bandwidth
Double-sided bandwidth

49. double-sideband bandwidth
Baseband and double-sideband bandwidth
Local oscillator
double-sideband bandwidth
Baseband
bandwidth

50. Spectrum of Raised Cosine Pulse
r=0 corresponds to sinc(.) function
1.0
0.5
f (Hz)

51. Raised Cosine Pulse - Time Domain
Sync fn

52. Raised Cosine Pulse - Frequency Domain

53. Implementation of Raised Cosine Pulse
Can be digitally implemented with an FIR filter
Analog filters such as Butterworth filters may approximate the tight shape of this spectrum
Practical pulses must be truncated in time
Truncation leads to sidelobes - even in RC pulses
Sometimes a “square-root” raised cosine spectrum is used when identical filters are implemented at transmitter and receiver
We will discuss this more for “matched filtering.”

54. Bandwidth of Raised Cosine Pulses
For PCM system: 2n=L (1 sample = n bits)
is a parameter called “roll-off factor”
Special cases:
r = 0 is just an Sa(.) function
r = 1 is the largest possible value
r = 0.35 is used in U.S. Digital Cellular (IS-54/136) standard
r = 0.22 is used in WCDMA (3G) standard
Sampling rate
SSB

55. Raised – Cosine Filter ( R-C Filter ) (3)
(i) When r = 1, the required excess bandwidth is
100% and the tails are quite small.
(ii) Bandpass - modulated signals ( chap. 4 ), such as
Amplitude-Shift Keying (ASK) and Phase-Shift
Keying (PSK), require twice the transmission
bandwidth of the equivalent baseband signals. WDSB
(iii) The larger the filter roll-off, the shorter will be
the pulse tails ( which implies smaller tail amplitudes ).

56. Raised – Cosine Filter ( R-C Filter ) (4)
large r  Small tail’s exhibit less sensitivity to timing error’s and thus make for small degradation due to ISI.
(v) The smaller the filter roll-off is,
the smaller the excess BW will be.
 Increase the signaling rate or the number of users
that can simultaneously use system.
 The greater sensitivity to timing errors.

57. Recall our dilemma!! Infinite bandwidth FT f t r=0 FT t f Inter-symbol
Infinite duration
and non-causal
r=0 FT t f
Inter-symbol
interference
양쪽에서 조금씩 타협

58. Raised Cosine (R-C) Filter ( 2 - ASK , r = 0 )
Roll-off Factor : r = 0
Filter Duration = [-4Ts,4Ts]
No. of Oversampling = 8
[8 samples/Ts]
Large ISI
Output of Matched Filter
[1,-1,-1,1,-1,1,1,-1,1,-1]

59. Raised Cosine Filter ( 2 - ASK , r = 0.5 )
Roll-off Factor : r = 0.5
Filter Duration = [-4Ts,4Ts]
No. of Oversampling = 8
Output of Matched Filter
[1,-1,-1,1,-1,1,1,-1,1,-1]

60. Raised Cosine Filter ( 2 - ASK , r = 1 )
Roll-off Factor : r = 1.0
Filter Duration = [-4Ts,4Ts]
No. of Oversampling = 8
Small ISI

61. 3.3.2 Two types of Error-performance Degradation
Without ISI –
theoretical
practical
With ISI
– More may not help the ISI problem.
– Equalization will help.

62. Example of Bandwidth Requirements (1)
Example (a) Find minimum required bandwidth for
4-ary PAM baseband modulation (R = 2400 bps , r = 1)
Find minimum required bandwidth for 4-ary PAM
bandpass modulation
( Answer ) (a) M = 4  k = 2 bits
symbol rate
(b)

63. Example of Bandwidth Requirements (2)
Ex 3.4) Digital Telephone Circuits ( PCM Waveform )
– 3 kHz analog voice  8 bit ADC  fs = 8kHz
– Bit rate : R = 8000  8 bits = 64 kHz
– For ideal Nyquist Filtering ,
Note) Binary signaling with a PCM waveform requires at least eight times the BW required for the analog channel .  needs speech codec, 10kbps
(analog BW : guard-band + 3 kHz = 4 kHz )

64. 3.3.3 Demodulation/Detection of shaped pulse
Conventional : filter-out unwanted spectrum
Matched : maximize energy at the sample points T.
Optimal under AWGN
Sample point
Nyquist pulse
with ISI * = t t t 2T T
Sample point
No ISI = * t t t T

65. 3.3.3 Demodulation/Detection of shaped pulse
Nyquist waveform observation
The square-root raised cosine filter  non zero ISI
transmitter output  not exact original samples
MF(matched filter) output  zero ISI at the sample points.
[ ]
[ ]
Channel
correlator
transmitter output
Matched filter output

66. Eye Pattern  Oscilloscope display for the received signal
on the vertical input with the horizontal sweep
rate 1/Ts
 The optimum sampling time corresponds to
the maximum eye opening , yielding the greatest
protection against noise .
 If there were no filtering in the system , then
the system response would yield ideal rectangular
DA : Measure of distortion caused by ISI
JT : Measure of the timing jitter
MN : Measure of noise margin
ST : Sensitivity-to timing error
pulse shapes . ( no filtering  BW corresponding
to the transmission of the data pulse is infinite )
 As the eye closes , ISI is increasing ; as the eye
opens , ISI is decreasing .

67. Practical Eye Pattern for ASK (or PAM)

68. Type of Eye Pattern

69. Eye Pattern ( 4 – ASK ) with Roll-off Factor