DIGITAL SPREAD SPECTRUM SYSTEMS

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DIGITAL SPREAD SPECTRUM SYSTEMS
ENG-737 Lecture 1

• Wright State University
• James P. Stephens

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CONTACT INFORMATION

• Email james.stephens_at_wpafb.af.mil
• Phone [phone]
• Web Page http//www.cs.wright.edu/jstephen/
• Fax [phone]

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COURSE SCHEDULE
WK 1 Introduction, digital comm. fundamentals
WK 2 Fourier transforms, random signals, dimensionality
WK 3 PN sequences, code properties, characteristic equations
WK 4 Auto/Cross correlation, product codes
WK 5 Direct sequence systems
WK 6 Frequency hopping systems
WK 7 Processing gain, jamming margin, jamming strategies
WK 8 Research topic presentations by students
WK 9 Low probability of detection, CDMA, system performance, bit-error rate, quadrature signals
WK 10 Applications and future trends such as cellular radio, UWB, transform domain, multicarrier, OFDM, 802.11
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COURSE SCHEDULE
Week 1 1,3 April
Week 2 8,10 April
Week 3 15,17 April
Week 4 22, 24 April
Week 5 29 April, 1 May
Week 6 6,8 May - Quiz 1
Week 7 13,15 May
Week 8 20,22 May - Topics
Week 9 27,29 May
Week 10 3,5 June
Finals 9-14 June - Final
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WHAT WILL YOU LEARN ?

1. An overview of modern communications issues and
   trends
2. The definition of SS and a variety of
   applications
3. How to distinguish between various digital
   modulation types
4. An understanding of Spread Spectrum Anti Jam (AJ)
   / Low Probability-of-Intercept (LPI) waveforms
5. The concept of signal dimensionality and how it
   relates to processing gain and jamming margin
6. The properties of pseudonoise (PN) sequences, how
   to generate and exploit them
7. The function of each SS subsystem, i.e.
   modulation, demodulation, acquisition, tracking,
   etc
8. Optimal countermeasure techniques against SS
   communication systems
9. The concept of bit-error rate (BER) and how to
   determine system performance
10. New areas of digital comm. concepts such as
    cognitive radio, OFDM, multi-carrier CDMA, UWB,
    and AdHoc nets

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MODERN COMMUNICATIONS TRENDS
PSK
QAM
FSK
CODING
Technologies
Software Radio
Cognitive Systems
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COMMUNICATION SYSTEM BLOCK DIAGRAM
Carrier signal
?
s(t)
v(t)
r(t)
s(t)
message
n(t)

• Message Signal - Information signal or baseband
  signal
• Transmitter Converts and transmits message
  signal
• Baseband converter Filtering, encoding, and
  multiplexing
• Carrier wave modulation
• Power Amplification
• Channel Hard-wired or free space medium over
  which signal is transmitted (Signal is corrupted
  with noise and distorted)
• Receiver Demodulates, demultiplexes, and
  decodes received signal
• Received Signal Detected signal in the presence
  of noise is only an estimate of the message signal

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BLOCK DIAGRAM OF TYPICAL DIGITAL COMMUNICATION
SYSTEM
Source Introduction to Spread Spectrum
Communications by Peterson, Ziemer, and Borth
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GENERIC DIGITAL COMMUNICATION SYSTEM
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INFORMATION RATE vs. SYMBOL RATE

• Information in its most fundamental form is
  measured in bits (binary digits)
• A signal which conveys binary information is the
  binary digital waveform

Rs Symbol (baud) rate 1/T symbols / sec Rb
Information rate 1/T bits / sec Example Rs
1000 symbols / sec Rb 1000 bits / sec
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INFORMATION RATE vs. SYMBOL RATE (Cont.)
Binary information can be conveyed by M-ary
(multilevel) digital waveforms
Tb
Rs symbol (baud) rate 1/T symbols/sec Rb
information rate (1/T) Log2 M 1/Tb Where M
of levels in the M-ary waveform M
2k 23 8 k number of bits For the example
shown Rs 333 symbols/sec, Rb 1000 bits/sec
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Digital Encoding
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Digital Decoding
RD , PB Bits / Sec
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COMM SYSTEM PERFORMANCE PARAMETERSAnalog Systems
Performance Measure Signal-to-Noise Ratio (SNR)
2
1
4
3
Modulator
Demod
Channel

1. Analog Message (tone)
2. AM Modulator Output
3. Received RF Signal
4. Demodulated Output

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COMM SYSTEM PERFORMANCE PARAMETERSDigital Systems
Performance Measure Bit Error Rate (BER)
1
2
3
4
5
6
Filter
Decision
Filter
Demod
Mod
Channel

1. Digital Message
2. Filter Output
3. Modulator Output (ASK)
4. Demodulator Output (with noise)
5. Receiver Filter Output
6. Detector Output

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WHY DIGITAL?
But let your communication be, Yea, yea Nay,
nay for whatever is more than these, cometh of
evil. - The Gospel According to St.
Matthew (537)

• Demand Increased requirement for computer to
  computer communications
• Signal Regeneration Resistant to noise,
  interference, and distortion
• Digital Signal Processing Allows error detection
  and correction permits ease of encryption and
  use of AJ techniques
• Technology Digital communication systems are
  more flexible and are better suited for future
  communication needs
• Economics Components are becoming increasingly
  more available and less expensive. Ability to
  rapidly troubleshoot and repair systems reduce
  overall maintenance costs as well as increasing
  system availability

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PERFORMANCE CRITERIADigital vs. Analog Systems

• Analog communication systems reproduce waveforms
  (an infinite set)
• A figure of merit for analog systems is a
  fidelity criterion (e.g. percent distortion
  between transmitted and recovered waveforms)
• Digital communication systems transmit waveforms
  that represent digits (a finite set)
• A figure of merit for digital systems is the
  quantity of incorrectly detected digits

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FOURIER ANALYSIS

• Fourier analysis techniques are very useful for
• Determining the distribution of signal energy in
  the frequency domain
• Determining the output of a linear system given
  its input
• For Example

Output
Input
Linear System
Linear System
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FOURIER ANALYSIS

• FOURIER SERIES
• s(t) ?Fn e jn?t
• Where,
• Fn 1/T ? f(t) e jn?t dt
• FOURIER TRANSFORM
• F(?) ? f(t) e j?t dt
• f(t) ? F(?) e j?t d?

?
n - ?
Periodic Signals
T
?
- ?
Non Periodic Signals
?
- ?
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FOURIER ANALYSIS (Cont.)
Non-periodic Continuous Time
t
t
n
Periodic Discrete Time
f(n)
Fourier Series
k
n
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FOURIER SERIES OF A PULSE TRAIN
F
F
F
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IMPORTANT FOURIER PROPERTIES

• Scaling in time and frequency
• if f(t) F?
• then Ff(t/?) ? f(t/?) e-j?tdt
• Changing the variables by letting x t/?
• Ff(t/?) ? ? f(x) e-j(??)xdx
• ? F??

?
- ?
?
- ?
We see that if the time scale is reduced by a
factor of ? , the frequency scale is increased by
?.
f (t/?) ? F??
Ex. If the pulse width in time is reduced from
msec to ?sec (factor of 1000), the bandwidth
increases from kHz to MHz (a factor of 1000).
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IMPORTANT FOURIER PROPERTIES

• Frequency Translation
• If F? is shifted to the right by ?
  radians/sec, then f(t) is multiplied by e-j? t
• Proof F? ?0 ? f(t) e-j(? ? ) dt
• ? f(t)
  e j? t e-j?t dt
• or F? ?0 f(t) e j? t
• F? ?0 f(t) e -j? t

0
?
0
- ?
?
0
- ?
0
0
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IMPORTANT FOURIER PROPERTIES

• Example of Frequency Translation
• g(t) f(t) cos ?0 t
• cos ?0 t (1/2)(e j? t e -j? t)
• Then, Fg(t) F(f(t)/2) e j? t
  F(f(t)/2) e -j? t

0
0
0
0
Now applying the frequency translation property
we observe that the first term on the right-hand
side of the above equation is F(f(t)/2) e j?
t ½ F(? ?o) Which is F(?) shifted to the
right on the frequency axis and scaled by a
factor of 1/2
0
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IMPORTANT FOURIER PROPERTIES

• The previous example is called the modulation
  property because it is used to translate signals
  from baseband to RF for transmission purposes

F(?-? )
0
?
?0
?0B
?0-B
-?0B
-?0
-?0-B
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LINEAR SYSTEMS

• Linear systems are characterized by their
• Impulse response, h(t) time domain
• Frequency response, H(?) frequency domain
• The concept of impulse response

Fourier Transform of h(t)

• The concept of frequency response
• if x(t) e j?t ? a complex sinusoid
• then y(t) ?h(?) e j?(t - ?)d? ej?t F h(t)
• y(t) e j?t H(?) ? the system
  frequency response

?
- ?
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SIGNALS, CIRCUITS, AND SPECTRA

• Frequency spectra can be ascribed to both signals
  and circuits
• Passing a signal x(t) through a filtering circuit
  yields g(t) in the time domain, or G(?) in the
  frequency domain
• where g(t) x(t) h(t)
• and G(?) X(?) H(?)
• The output bandwidth is always constrained by the
  smaller of the two bandwidths

INPUT SIGNAL SPECTRUM
FILTER TRANSFER FUNCTION
CASE 1
Output bandwidth is constrained by input signal
bandwidth
CASE 2
Output bandwidth is constrained by filter
bandwidth
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GRAPHICAL EXAMPLE OF CONVOLUTION
Joy of Convolution http//www.jhu.edu/signals/co
nvolve/
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FILTERED PULSE-SIGNAL EXAMPLES
x(t)
Vm
R
Input
Low Pass Filter
C
t
H(f)
Input Pulse
Filter Frequency Response (Transfer Function)
1
0.707
f
f
1/2pRC
0
0
1/T
Wp ltlt Wf (T gtgt 2pRC)
t
Wp Wf (T 2pRC)
t
Wp gtgtWf (T ltlt 2pRC)
t
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RANDOM SIGNALS

• Fourier Analysis as previously described applies
  only to deterministic signals.
• A random signal (random process) appears as
  follows

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RANDOM SIGNALS

• The two most important random signals we
  encounter in digital communications are
• Noise
• Random binary waveforms
• Noise that is completely random is called white
  noise

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RANDOM SIGNALS (Cont)

• The noise encountered most in communications has
  a Gaussian amplitude distribution

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RANDOM SIGNALS Random Signals can only be
described statistically
T/2
Ex(t) lim 1/T ? x(t)dt
DC Value of signal
Signal Mean
T? 8
-T/2
Signal Autocorrelation
T/2
Ex2(t) lim 1/T ? x2(t)dt
Total signal power
T? 8
-T/2
Signal Variance
?x2 E(x Ex)2 Signal AC power
Ex2(t) ?x2 E2x(t) Total power AC power
DC power
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AUTOCORRELATION FUNCTION

• Provides a measure of the similarity of a signal
  with a time-delayed version of itself
• The autocorrelation function of a real-valued
  energy signal x(t) is defined as
• Rx(?) ? x(t) x(t ?) dt
• Important properties of Rx(?) are
• Symmetrical in ? about zero
• Maximum value occurs at the origin
• Autocorrelation and PSD form a Fourier transform
  pair
• Value at the origin is equal to the average power
  of the signal

?
- ?
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AUTOCORRELATION AND POWER SPECTRAL DENSITY
LOW BIT RATE
HIGH BIT RATE
x(t) Random binary sequence
x(t ?)
R(?) Ex(t) x(t ?)
1 - ?/T for ? lt T
R(?)
0 for ? gt T
8
-8
S(f) T sin (p/T) / p/T
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DISCRETE AUTOCORRELATION

• Spectral analysis of random processes differs
  from that of deterministic signals
• For stationary random processes, the
  autocorrelation function Rxx(?) tells us
  something about how rapidly we can expect the
  random signal to change as a function of time
• The autocorrelation for discrete time-series
  signal is defined for real signal xn

• If the autocorrelation function decays
  rapidly to zero it indicates that the process can
  be expected to change rapidly with time
• A slowly changing process will have an
  autocorrelation function that decays slowing
• If the autocorrelation function has periodic
  components, then the underlying process will also
  have periodic components

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DISCRETE CROSSCORRELATION

• The crosscorrelation for discrete time-series
  signals is defined for real signal xn

• The crosscorrelation function is even
• Another important property

• This is useful for time-of-arrival (TOA)
  measurements (ranging)
• Send out xn
• Cross correlate xn with the delayed return
  signal
• Peak in crosscorrelation will occur at n0 which
  is the amount of time delay

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ERROR-FREE CAPACITY

• It is useful to explore briefly the concept of
  capacity of a digital communications link
• Given the constraints of Power, Bandwidth, and
  AWGN, there exists a maximum rate at which
  information can be transmitted with high
  reliability (This rate is called error-free
  capacity)
• Pioneering work by Claude Shannon in the late
  1940s found
• C W log2 (1 P/N0W)
• C W log2 (1 Eb/N0(R/W))
• Where,
• C channel capacity in bits/sec
• W transmission bandwidth in Hz
• P EbR received signal power in watts
• N0 single-sided noise power density in
  watts/Hz
• Eb energy per bit of received signal
• R information rate in bits per signal
• C is the maximum capacity at which information
  can be put through the channel
• The goal is to make the information rate less
  than C in order to have reliable communications

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SHANNONS 2ND THEOREM

• Channel capacity may also be written
• C W log2 (1 S/N)
• Where,
• N N0W
• S EbR
• Spread spectrum systems typically operate with
  S/N lt 1 (i.e. more noise than signal due to the
  wide bandwidth used)

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SHANNONS 2ND THEOREM

• We can derive the significance of this law as
  follows
• Change the logarithm base
• log2 / loge 1.44, (i.e. log2(2) / ln(2)
  1.44)
• Therefore,
• C/W 1.44 loge (1 S/N)
• Since S/N lt0.1 for spread spectrum, loge(1 S/N)
  ? 1.44(S/N)
• (You can verify this on your calculator)
• C/W ? 1.44 (S/N) or C ? 1.44 W (S/N) (assuming
  low S/N)
• Significance For any given SNR, we get a low
  error rate by increasing the bandwidth used to
  transfer information
• Shannons limit is based upon the assumption that
  the noise is AWGN (Since noise is the only cause
  of errors, capacity is directly proportional to
  SNR)

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NYQUIST CHANNEL CAPACITY

• Another finding
• C(Nyquist) 2W log2 M
• Where,
• C is Nyquist Channel capacity in bits per sec
• M is the number of bits per symbol

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WHAT IS THE BANDWIDTH OF DIGITAL DATA ?
General shape of power spectral density (PSD)
S(t) random digital sequence
-1/T
1/T
BANDWIDTH CRITERIA
BW

• Half-Power
• Noise Equivalent
• Null-to-Null
• 99 of Power
• Bounded PSD
• - 35 dB
• - 50 dB

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BANDWIDTH DEFINITIONS
0
0
NOISE EQUIVALENT BANDWIDTH