Point-to-Point Wireless Communication (I): Digital Basics, Modulation, Detection, Pulse Shaping
Shiv Kalyanaraman Google: “Shiv RPI” [email protected]
Based upon slides of Sorour Falahati, A. Goldsmith, & textbooks by Bernard Sklar & A. Goldsmith Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 1 : “shiv rpi” The Basics
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 2 : “shiv rpi” Big Picture: Detection under AWGN
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 3 : “shiv rpi” Additive White Gaussian Noise (AWGN)
Thermal noise is described by a zero-mean Gaussian random process, n(t) that ADDS on to the signal => “additive” Its PSD is flat, hence, it is called white noise. Autocorrelation is a spike at 0: uncorrelated at any non-zero lag
[w/Hz]
Power spectral Density (flat => “white”)
Autocorrelation Function Probability density function (uncorrelated) (gaussian) Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 4 : “shiv rpi” Effect of Noise in Signal Space
The cloud falls off exponentially (gaussian). Vector viewpoint can be used in signal space, with a random noise vector w
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 5 : “shiv rpi” Maximum Likelihood (ML) Detection: Scalar Case
“likelihoods”
Assuming both symbols equally likely: uA is chosen if
Log-Likelihood => A simple distance criterion! Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 6 : “shiv rpi” AWGN Detection for Binary PAM
z z mp 2 )|( z z mp 1)|(
s2 s1 ψ 1 t)( 0 − Eb Eb
⎛ − ss 2/ ⎞ )()( == QmPmP ⎜ 21 ⎟ e 1 e 2 ⎜ ⎟ ⎝ N0 2/ ⎠ ⎛ 2E ⎞ )2( == QPP ⎜ b ⎟ EB ⎜ ⎟ ⎝ N0 ⎠ Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 7 : “shiv rpi” Bigger Picture General structure of a communication systems Noise Transmitted Received Received Info. signal signal info. SOSourceURCE Transmitter Channel Receiver User
Transmitter
Source Channel Formatter Modulator encoder encoder
Receiver
Source Channel Formatter Demodulator decoder decoder
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 8 : “shiv rpi” Digital vs Analog Comm: Basics
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 9 : “shiv rpi” Digital versus analog
Advantages of digital communications: Regenerator receiver
Original Regenerated pulse pulse
Propagation distance Different kinds of digital signal are treated identically. Voice Data A bit is a bit! Media
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 10 : “shiv rpi” Signal transmission through linear systems
Input Output Linear system Deterministic signals: Random signals:
Ideal distortion less transmission: All the frequency components of the signal not only arrive with an identical time delay, but also are amplified or attenuated equally.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 11 : “shiv rpi” Signal transmission (cont’d) Ideal filters:
Non-causal! Low-pass
Band-pass Duality => similar problemsHigh-pass occur w/ rectangular pulses in time domain.
Realizable filters: RC filters Butterworth filter
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 12 : “shiv rpi” Bandwidth of signal
Baseband versus bandpass:
Baseband Bandpass signal signal Local oscillator
Bandwidth dilemma: Bandlimited signals are not realizable! Realizable signals have infinite bandwidth!
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 13 : “shiv rpi” Bandwidth of signal: Approximations
Different definition of bandwidth: a) Half-power bandwidth d) Fractional power containment bandwidth b) Noise equivalent bandwidth e) Bounded power spectral density c) Null-to-null bandwidth f) Absolute bandwidth
(a) (b) (c) (d) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute (e)50dB 14 : “shiv rpi” Formatting and transmission of baseband signal
Digital info.
Textual Format source info. Pulse Transmit Analog Sample Quantize Encode modulate info.
Pulse Bit stream waveforms Channel Format Analog info. Low-pass Decode Demodulate/ filter Receive Detect sink Textual info.
Digital info.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 15 : “shiv rpi” Sampling of Analog Signals
Time domain Frequency domain = × txtxtx )()()( s δ s = δ ∗ fXfXfX )()()( tx )( fX |)(|
fX |)(| δ tx )( δ
s tx )( s fX |)(|
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 16 : “shiv rpi” Aliasing effect & Nyquist Rate
LP filter
Nyquist rate
aliasing
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 17 : “shiv rpi” Undersampling &Aliasing in Time Domain
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 18 : “shiv rpi” Nyquist Sampling & Reconstruction: Time Domain
Note: correct reconstruction does not draw straight lines between samples. Key: use sinc() pulses for reconstruction/interpolation Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 19 : “shiv rpi” Nyquist Reconstruction: Frequency Domain
The impulse response of the reconstruction filter has a classic 'sin(x)/x shape.
The stimulus fed to this filter is the series of discrete impulses which are the samples.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 20 : “shiv rpi” Sampling theorem
Analog Sampling Pulse amplitude signal process modulated (PAM) signal
Sampling theorem: A bandlimited signal with no spectral components beyond , can be uniquely determined by values sampled at uniform intervals of
The sampling rate,
is called Nyquist rate.
In practice need to sample faster than this because the receiving filter will not be sharp. Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 21 : “shiv rpi” Quantization
Amplitude quantizing: Mapping samples of a continuous amplitude waveform to a finite set of amplitudes.
Out
In Average quantization noise power
Signal peak power values
Quantized Signal power to average quantization noise power
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 22 : “shiv rpi” Encoding (PCM)
A uniform linear quantizer is called Pulse Code Modulation (PCM). Pulse code modulation (PCM): Encoding the quantized signals into a digital word (PCM word or codeword). Each quantized sample is digitally encoded into an l bits codeword where L in the number of quantization levels and
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 23 : “shiv rpi” Quantization error
Quantizing error: The difference between the input and output of a quantizer )( = ˆ − txtxte )()(
Process of quantizing noise Qauntizer Model of quantizing noise = xqy )(
AGC x(t) xˆ(t) tx )( ˆ tx )( x e(t)
+ te )( = ˆ − txtx )()(
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 24 : “shiv rpi” Non-uniform quantization
It is done by uniformly quantizing the “compressed” signal. At the receiver, an inverse compression characteristic, called “expansion” is employed to avoid signal distortion.
compression+expansion companding
= xCy )( xˆ tx )( ty )( ˆ ty )( ˆ tx )(
x yˆ Compress Qauntize Expand Transmitter Channel Receiver Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 25 : “shiv rpi” Baseband transmission
To transmit information thru physical channels, PCM sequences (codewords) are transformed to pulses (waveforms). Each waveform carries a symbol from a set of size M.
Each transmit symbol represents k = log 2 M bits of the PCM words. PCM waveforms (line codes) are used for binary symbols (M=2).
M-ary pulse modulation are used for non-binary symbols (M>2). Eg: M-ary PAM. For a given data rate, M-ary PAM (M>2) requires less bandwidth than binary PCM. For a given average pulse power, binary PCM is easier to detect than M- ary PAM (M>2).
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 26 : “shiv rpi” PAM example: Binary vs 8-ary
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 27 : “shiv rpi” Example of M-ary PAM
Assuming real time Tx and equal energy per Tx data bit for binary-PAM and 4-ary PAM:
• 4-ary: T=2Tb and Binary: T=Tb
• Energy per symbol in binary-PAM: 2 =10BA 2
Binary PAM 4-ary PAM (rectangular pulse) (rectangular pulse) 3B A. ‘11’ ‘1’ B T T T ‘01’ T T T -B ‘00’ ‘0’ ‘10’ -A. -3B Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 28 : “shiv rpi” Other PCM waveforms: Examples
PCM waveforms category:
Nonreturn-to-zero (NRZ) Phase encoded Return-to-zero (RZ) Multilevel binary
+V 1 0 1 1 0 +V 1 0 1 1 0 NRZ-L -V Manchester -V
Unipolar-RZ +V Miller +V 0 -V +V +V Bipolar-RZ 0 Dicode NRZ 0 -V -V 0 T 2T 3T 4T 5T 0 T 2T 3T 4T 5T
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 29 : “shiv rpi” PCM waveforms: Selection Criteria
Criteria for comparing and selecting PCM waveforms: Spectral characteristics (power spectral density and bandwidth efficiency) Bit synchronization capability Error detection capability Interference and noise immunity Implementation cost and complexity
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 30 : “shiv rpi” Summary: Baseband Formatting and transmission
Digital info. Bit stream Pulse waveforms (Data bits) (baseband signals) Textual Format source info. Pulse Analog Sample Quantize Encode modulate info.
Sampling at rate Encoding each q. value to = /1 Tf s s = log2 Ll bits (sampling time=Ts) (Data bit duration Tb=Ts/l)
Quantizing each sampled Mapping every = log 2 Mm data bits to a value to one of the symbol out of M symbols and transmitting L levels in quantizer. a baseband waveform with duration T Information (data- or bit-) rate: b = /1 TR b [bits/sec] Symbol rate : = /1 TR [symbols/s ec]
Rensselaer Polytechnic Institute b = mRR Shivkumar Kalyanaraman 31 : “shiv rpi” Receiver Structure & Matched Filter
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 32 : “shiv rpi” Demodulation and detection
m Pulse i tg )( Bandpass i ts )( Format i M-ary modulation modulate modulate = K,,1 Mi channel transmitted symbol c th )( estimated symbol tn )( Demod. Format Detect mˆ i Tz )( & sample tr )(
Major sources of errors: Thermal noise (AWGN) disturbs the signal in an additive fashion (Additive) has flat spectral density for all frequencies of interest (White) is modeled by Gaussian random process (Gaussian Noise) Inter-Symbol Interference (ISI) Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared”. Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 33 : “shiv rpi” Impact of AWGN
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 34 : “shiv rpi” Impact of AWGN & Channel Distortion
c δ δ −−= Tttth )75.0(5.0)()(
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 35 : “shiv rpi” Receiver job
Demodulation and sampling: Waveform recovery and preparing the received signal for detection: Improving the signal power to the noise power (SNR) using matched filter (project to signal space) Reducing ISI using equalizer (remove channel distortion) Sampling the recovered waveform
Detection: Estimate the transmitted symbol based on the received sample
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 36 : “shiv rpi” Receiver structure
Step 1 – waveform to sample transformation Step 2 – decision making
Demodulate & Sample Detect
Tz )( Threshold mˆ tr )( Frequency Receiving Equalizing i comparison down-conversion filter filter
For bandpass signals Compensation for channel induced ISI
Baseband pulse Received waveform Sample (possibly distorted) Baseband pulse (test statistic)
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 37 : “shiv rpi” Baseband vs Bandpass
Bandpass model of detection process is equivalent to baseband model because: The received bandpass waveform is first transformed to a baseband waveform.
Equivalence theorem: Performing bandpass linear signal processing followed by heterodying the signal to the baseband, … … yields the same results as … … heterodying the bandpass signal to the baseband , followed by a baseband linear signal processing.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 38 : “shiv rpi” Steps in designing the receiver
Find optimum solution for receiver design with the following goals: 1. Maximize SNR: matched filter 2. Minimize ISI: equalizer
Steps in design: Model the received signal Find separate solutions for each of the goals.
First, we focus on designing a receiver which maximizes the SNR: matched filter
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 39 : “shiv rpi” Receiver filter to maximize the SNR
Model the received signal
ts )( tr )( i c th )( = ∗ ci + tnthtstr )()()()(
tn )( AWGN Simplify the model (ideal channel assumption): Received signal in AWGN
Ideal channels tr )( i ts )( = i + tntstr )()()( c = δ tth )()(
tn )( AWGN
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 40 : “shiv rpi” Matched Filter Receiver Problem: Design the receiver filter th )( such that the SNR is
maximized at the sampling time when i ),( = ,...,1 Mits is transmitted. Solution: The optimum filter, is the Matched filter, given by * opt i −== tTsthth )()()( * = opt = i − πfTjfSfHfH )2exp()()()(
which is the time-reversed and delayed version of the conjugate of the transmitted signal
i ts )( = opt thth )()(
0tT 0tT Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 41 : “shiv rpi” Correlator Receiver The matched filter output at the sampling time, can be realized as the correlator output. * Matched filtering, i.e. convolution with si (T-τ) simplifies to * integration w/ si (τ), i.e. correlation or inner product!
= opt ∗ TrThTz )()()( T = * τττ=< tstrdsr )(),()()( > ∫ i 0
Recall: correlation operation is the projection of the received signal onto the signal space!
Key idea: Reject the noise (N) outside this space as irrelevant: => maximize S/N Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 42 : “shiv rpi” Irrelevance Theorem: Noise Outside Signal Space
Noise PSD is flat (“white”) => total noise power infinite across the spectrum. We care only about the noise projected in the finite signal dimensions (eg: the bandwidth of interest).
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 43 : “shiv rpi” Aside: Correlation Effect
Correlation is a maximum when two signals are similar in shape, and are in phase (or 'unshifted' with respect to each other). Correlation is a measure of the similarity between two signals as a function of time shift (“lag”, τ ) between them When the two signals are similar in shape and unshifted with respect to each other, their product is all positive. This is like constructive interference, The breadth of the correlation function - where it has significant value - shows for how long the signals remain similar. Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 44 : “shiv rpi” Aside: Autocorrelation
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 45 : “shiv rpi” Aside: Cross-Correlation &Radar
Figure: shows how the signal can be located within the noise. A copy of the known reference signal is correlated with the unknown signal. The correlation will be high when the reference is similar to the unknown signal. A large value for correlation shows the degree of confidence that the reference signal is detected. The large value of the correlation indicates when the reference signal occurs.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 46 : “shiv rpi” • A copy of a known reference signal is correlated with the unknown signal. • The correlation will be high if the reference is similar to the unknown signal. • The unknown signal is correlated with a number of known reference functions. • A large value for correlation shows the degree of similarity to the reference. • The largest value for correlation is the most likely match.
• Same principle in communications: reference signals corresponding to symbols • The ideal communications channel may have attenuated, phase shifted the reference signal, and added noise
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 47 Source: Bores Signal: “shiv Processing rpi” Matched Filter: back to cartoon…
Consider the received signal as a vector r, and the transmitted signal vector as s Matched filter “projects” the r onto signal space spanned by s (“matches” it)
Filtered signal can now be safely sampled by the receiver at the correct sampling instants, resulting in a correct interpretation of the binary message
Matched filter is the filter that maximizes the signal-to-noise ratio it can be shown that it also minimizes the BER: it is a simple projection operation
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 48 : “shiv rpi” Example of matched filter (real signals)
= i ∗ opt thtsty )()()( 2 i ts )( opt th )( A A A T T
Tt Tt 02TtT
= i ∗ opt thtsty )()()( 2 i ts )( opt th )( A A A T T
T/2 Tt T/2 TtT 02T/2Tt 3T/2 T 2 − A − A A − 2 T T
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 49 : “shiv rpi” Properties of the Matched Filter 1. The Fourier transform of a matched filter output with the matched signal as input is, except for a time delay factor, proportional to the ESD of the input signal. = 2 − πfTjfSfZ )2exp(|)(|)( 2. The output signal of a matched filter is proportional to a shifted version of the autocorrelation function of the input signal to which the filter is matched.
= s − ⇒ = s )0()()()( = ERTzTtRtz s
3. The output SNR of a matched filter depends only on the ratio of the signal energy to the PSD of the white noise at the filter input. ⎛ S ⎞ E max⎜ ⎟ = s ⎝ N ⎠T N0 2/ 4. Two matching conditions in the matched-filtering operation: spectral phase matching that gives the desired output peak at time T. spectral amplitude matching that gives optimum SNR to the peak value.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 50 : “shiv rpi” Implementation of matched filter receiver
Bank of M matched filters
1 Tz )( * − tTs )( 1 ⎡ z1 ⎤ Matched filter output: tr )( z ⎢ ⎥ = z Observation ⎢ M ⎥ vector
* ⎢z ⎥ − tTs )( ⎣ M ⎦ M M Tz )(
∗ i i −∗= tTstrz )()( = ,...,1 Mi
z = 21 M = 21 zzzTzTzTz M ),...,,())(),...,(),((
Note: we are projecting along the basis directions of the signal space Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 51 : “shiv rpi” Implementation of correlator receiver Bank of M correlators
∗ 1 ts )(
T 1 Tz )( ∫0 ⎡ z1 ⎤ Correlators output: tr )( ⎢ ⎥ z Observation ∗ = z vector M ts )( ⎢ M ⎥ T ⎣⎢zM ⎦⎥ ∫0 M Tz )(
z = 21 M = 21 zzzTzTzTz M ),...,,())(),...,(),(( T = )()( dttstrz = ,...,1 Mi i ∫ i 0
Rensselaer PolytechnicNote: In Institute previous slide we “filter” i.e. convolute inShivkumar the boxes Kalyanaraman shown. 52 : “shiv rpi” Implementation example of matched filter receivers
1 ts )( A Bank of 2 matched filters T
0 Tt A 1 Tz )( T ⎡z1⎤ tr )( z 0 T ⎢ ⎥ = z
2 ts )( ⎢ ⎥ ⎣⎢z2⎦⎥ 0 T Tz )( 0 Tt 2 − A − A T T
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 53 : “shiv rpi” Matched Filter: Frequency domain View
Simple Bandpass Filter: excludes noise, but misses some signal power
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 54 : “shiv rpi” Matched Filter: Frequency Domain View (Contd) Multi-Bandpass Filter: includes more signal power, but adds more noise also!
Matched Filter: includes more signal power, weighted according to size => maximal noise rejection!
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 55 : “shiv rpi” Maximal Ratio Combining (MRC) viewpoint
Generalization of this f-domain picture, for combining multi-tap signal
Weight each branch
SNR:
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 56 : “shiv rpi” Examples of matched filter output for bandpass modulation schemes
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 57 : “shiv rpi” Signal Space Concepts
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 58 : “shiv rpi” Signal space: Overview
What is a signal space? Vector representations of signals in an N-dimensional orthogonal space
Why do we need a signal space? It is a means to convert signals to vectors and vice versa. It is a means to calculate signals energy and Euclidean distances between signals.
Why are we interested in Euclidean distances between signals? For detection purposes: The received signal is transformed to a received vectors. The signal which has the minimum distance to the received signal is estimated as the transmitted signal.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 59 : “shiv rpi” Schematic example of a signal space
ψ 2 t)(
s = aa 12111 ),(
ψ 1 t)(
z = zz 21 ),(
s = aa 32313 ),(
s = aa 22212 ),(
1 = ψ 111 + ψ 212 ⇔ s = aatatats 12111 ),()()()( Transmitted signal += ψψ s =⇔ aatatats ),()()()( alternatives 2 121 222 22212 3 131 += ψψ232 s =⇔ aatatats 32313 ),()()()( Received signal at 11 ψψ22 z =⇔+= zztztztz 21 ),()()()( matchedRensselaer filter Polytechnic output Institute Shivkumar Kalyanaraman 60 : “shiv rpi” Signal space
To form a signal space, first we need to know the inner product between two signals (functions): Inner (scalar) product: ∞ < >= ∫ * )()()(),( dttytxtytx ∞− = cross-correlation between x(t) and y(t)
Properties of inner product: < >= < tytxatytax )(),()(),( > < * <>= tytxataytx )(),()(),( > +< >=< > + < tztytztxtztytx )(),()(),()(),()( > Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 61 : “shiv rpi” Signal space …
The distance in signal space is measure by calculating the norm. What is norm? Norm of a signal:
∞ 2 txtxtx )(),()( =><= )( = Edttx x ∫ ∞− = “length”of x(t) = txatax )()(
Norm between two signals:
, yx −= tytxd )()(
We refer to the norm between two signals as the Euclidean distance between two signals. Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 62 : “shiv rpi” Example of distances in signal space
ψ 2 t)(
s = aa 12111 ),(
E1 d 1 ,zs
ψ 1 t)( E 3 z = zz 21 ),( d 3 ,zs E2 d 2 ,zs s = aa 32313 ),(
s = aa 22212 ),(
The Euclidean distance between signals z(t) and s(t): 2 −+−=−= zazatztsd )()()()( 2 i , izs i 11 i 22 i = 3,2,1
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 63 : “shiv rpi” Orthogonal signal space
N-dimensional orthogonal signal space is characterized by N {ψ t)( }N linearly independent functions j j= 1 called basis functions. The basis functions must satisfy the orthogonality condition
T 0 ≤ ≤ Tt < >= ψψψψ* )()()(),( = Kdttttt δ ji ∫ i j jii 0 = ,...,1, Nij where ⎧1 =→ ji δij = ⎨ ⎩0 ≠→ ji
If all K i = 1 , the signal space is orthonormal. Constructing Orthonormal basis from non-orthonormal set of vectors: Gram-Schmidt procedure
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 64 : “shiv rpi” Example of an orthonormal bases Example: 2-dimensional orthonormal signal space ⎧ 2 ⎪ψ 1 t)( = π 0)/2cos( <≤ TtTt ψ 2 t)( ⎪ T ⎨ ⎪ 2 ψ 2 t)( = π 0)/2sin( <≤ TtTt ⎩⎪ T ψ t)( T 0 1 < >= ψψψψdttttt = 0)()()(),( 21 ∫ 1 2 0
1 ψψ2 tt == 1)()(
Example: 1-dimensional orthonornal signal space
ψ 1 t)( 1 ψ 1 t =1)( T ψ t)( 0 1
0 Tt Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 65 : “shiv rpi” Sine/Cosine Bases: Note! Approximately orthonormal!
These are the in-phase & quadrature-phase dimensions of complex baseband equivalent representations.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 66 : “shiv rpi” Example: BPSK
Note: two symbols, but only one dimension in BPSK.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 67 : “shiv rpi” Signal space … M Any arbitrary finite set of waveforms {}i ts )( i=1 where each member of the set is of duration T, can be expressed as a linear combination of N orthogonal waveforms where {} ψ . t)( N ≤ MN j j=1
N = ,...,1 Mi i = ∑ ψ jij tats )()( j=1 ≤ MN where 1 1 T = ,...,1 Nj a <= ψ tts )(),( >= ψ * )()( dttts ij ji ∫ ji 0 ≤ ≤ Tt K j K j 0 = ,...,1 Mi
N 2 s = iii 21 aaa iN ),...,,( i = ∑ j aKE ij j=1 Vector representation of waveform Waveform energy Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 68 : “shiv rpi” Signal space …
N )( = ψ tats )( i ∑ jij s = iii 21 aaa iN ),...,,( j =1 Waveform to vector conversion Vector to waveform conversion ψ t)( 1 ψ 1 t)( T ai1 ai1 ⎡a ⎤ ⎡ai1 ⎤ ∫0 i1 s i ts )( m ts )( ⎢ ⎥ = s ⎢ ⎥ i ψ t)( m M N ⎢ M ⎥ ⎢ ⎥ ψ N t)( T ⎢a ⎥ ⎣⎢aiN⎦⎥ ⎣ iN⎦ ∫0 aiN aiN
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 69 : “shiv rpi” Example of projecting signals to an orthonormal signal space ψ 2 t)(
s = aa 12111 ),(
ψ 1 t)(
s = aa 32313 ),(
s = aa 22212 ),(
= ψ + ψ ⇔ s = aatatats ),()()()( Transmitted signal 1 111 212 12111 alternatives 2 121 += ψψ222 s =⇔ aatatats 22212 ),()()()(
3 131 += ψψ232 s =⇔ aatatats 32313 ),()()()( T = ψ )()( dtttsa = ,...,1 Nj Shivkumar Kalyanaraman Rensselaer Polytechnic Institute ∫ jiij = ,...,1 Mi 0 ≤ ≤ Tt 0 70 : “shiv rpi” Matched filter receiver (revisited) (note: we match to the basis directions) Bank of N matched filters
∗ z1 ψ 1 − tT )( Observation ⎡z1 ⎤ vector tr )( z ⎢ ⎥ = z ⎢ ⎥ ∗ ⎢z ⎥ ψ N − tT )( ⎣ N⎦ zN
N i = ∑ ψ jij tats )()( = ,...,1 Mi j=1 z = 21 zzz N ),...,,( ≤ MN
j ∗= ψ j − tTtrz )()( = ,...,1 Nj
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 71 : “shiv rpi” Correlator receiver (revisited)
Bank of N correlators
ψ 1 t)(
T z1 ∫0 ⎡r1 ⎤ tr )( z ⎢ ⎥ = z Observation ψ t)( vector N ⎢ M ⎥ T ⎣⎢rN ⎦⎥ ∫0 zN N i = ∑ ψ jij tats )()( = ,...,1 Mi j=1
z = 21 zzz N ),...,,( ≤ MN T = ψ )()( dtttrz = ,...,1 Nj j ∫ j Rensselaer Polytechnic0 Institute Shivkumar Kalyanaraman 72 : “shiv rpi” Example of matched filter receivers using basic functions
1 ts )( 2 ts )( ψ 1 t)( A 1 T T 0 Tt 0 Tt − A 0 Tt T 1 matched filter
ψ 1 t)( tr )( 1 z z T 1 = z []z1 0 Tt
Number of matched filters (or correlators) is reduced by 1 compared to using matched filters (correlators) to the transmitted signal! Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 73 : “shiv rpi” White noise in Orthonormal Signal Space
AWGN n(t) can be expressed as )( = ˆ )( + ~ tntntn )(
Noise projected on the signal space Noise outside on the signal space (colored):impacts the detection process. (irrelevant)
N ˆ = ψ tntn )()( ∑ jj Vector representation of ˆ tn )( j=1 n = nnn ),...,,( j =< ψ j ttnn )(),( > = ,...,1 Nj 21 N N ~ {}n independent zero-mean < ψ ttn >= 0)(),( = ,...,1 Nj j j=1 j Gaussain random variables with
variance j = Nn 0 2/)var(
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 74 : “shiv rpi” Detection: Maximum Likelihood & Performance Bounds
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 75 : “shiv rpi” Detection of signal in AWGN
Detection problem: Given the observation vector zz , perform a mapping from
to an estimate m ˆ of the transmitted symbol, m i , such that the average probability of error in the decision is minimized.
n
si z mi Modulator Decision rule mˆ
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 76 : “shiv rpi” Statistics of the observation Vector
AWGN channel model: = i + nsz
Signal vector s = iii 21 aaa iN ),...,,( is deterministic.
Elements of noise vector n = 21 nnn N ),...,,( are i.i.d Gaussian
random variables with zero-mean and variance N 0 2/ . The noise vector pdf is 2 1 ⎛ n ⎞ p n)( = exp⎜− ⎟ n ()πN N 2/ ⎜ N ⎟ 0 ⎝ 0 ⎠ The elements of observed vector z = 21 zzz N ),...,,( are independent Gaussian random variables. Its pdf is 2 1 ⎛ − sz ⎞ p sz )|( = exp⎜− i ⎟ z i ()πN N 2/ ⎜ N ⎟ 0 ⎝ 0 ⎠
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 77 : “shiv rpi” Detection
Optimum decision rule (maximum a posteriori probability):
Set ˆ = mm i if
allfor ,)|sent Pr()|sent Pr(mi z ≥ Pr()|sent mk z ,)|sent allfor ≠ ik .,...,1 where = Mk .,...,1
Applying Bayes’ rule gives:
Set ˆ = mm i if
z z mp k )|( pk , is maximum allfor = ik pz z)(
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 78 : “shiv rpi” Detection …
Partition the signal space into M decision regions,
such that 1,..., ZZ M
if region inside lies Vector z lies inside region Zi if
z z mp k )|( ln[ pk ], is maximum allfor = ik . pz z)( meansThat meansThat
ˆ = mm i
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 79 : “shiv rpi” Detection (ML rule)
For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to:
Set ˆ = mm i if
z z mp k ),|( is maximum allfor = ik
or equivalently:
Set ˆ = mm i if
z z mp k )],|(ln[ is maximum allfor = ik
which is known as maximum likelihood.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 80 : “shiv rpi” Detection (ML)…
Partition the signal space into M decision regions, 1,..., ZZ M Restate the maximum likelihood decision rule as follows:
if region inside lies Vector z lies inside region Zi if
z z mp k )],|(ln[ is maximum allfor = ik meansThat meansThat
ˆ = mm i
if region inside lies Vector z lies inside region Zi if
− sz k , is minimum allfor = ik
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 81 : “shiv rpi” Schematic example of ML decision regions
ψ 2 t)(
Z2
s2 Z s 1 3 s1 ψ 1 t)( Z3
s4
Z4 Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 82 : “shiv rpi” Probability of symbol error
Erroneous decision: For the transmitted symbol s i or equivalently signal vector m i , an error in decision occurs if the observation vector z does not fall inside region Z i. Probability of erroneous decision for a transmitted symbol
Pr( ˆ i =≠ Pr() mmm i sent)Pr( z does lienot inside mZ ii sent)
Probability of correct decision for a transmitted symbol
Pr( ˆ i == Pr() mmm i sent)Pr( z lies inside mZ ii sent)
mP = z liesPr()( inside mZ = )|(sent) dmp zz ic ii ∫ z i Zi ie = − mPmP ic )(1)(
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 83 : “shiv rpi” Example for binary PAM
z z mp 2 )|( z z mp 1)|(
s2 s1 ψ 1 t)( 0 − Eb Eb
⎛ − ss 2/ ⎞ )()( == QmPmP ⎜ 21 ⎟ e 1 e 2 ⎜ ⎟ ⎝ N0 2/ ⎠ ⎛ 2E ⎞ )2( == QPP ⎜ b ⎟ EB ⎜ ⎟ ⎝ N0 ⎠ Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 84 : “shiv rpi” Average prob. of symbol error …
Average probability of symbol error :
M ˆ E MP = ∑ (Pr)( ≠ mm i ) i =1
For equally probable symbols:
1 M 1 M E MP )( = ∑ mP ie 1)( −= ∑ mP ic )( M i=1 M i=1 1 M 1−= )|( dmp zz ∑ ∫ z i M i=1 Zi
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 85 : “shiv rpi” Union bound
Union bound The probability of a finite union of events is upper bounded by the sum of the probabilities of the individual events.
M 1 M M ie ≤ ∑ PmP 2 ss ik ),()( E MP )( ≤ ∑∑P2 ss ik ),( k =1 M i=11k= ≠ik ≠ik
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 86 : “shiv rpi” Example of union bound
ψ 2 Z2 r Z1 e 1 = r 1)|()( dmpmP rr s ∫ 2 s1 ∪∪ ZZZ 432 ψ 1
Union bound: s3 s4 4 Z 3 Z4 e 1 ≤ ∑ PmP k ss 12 ),()( k =2
ψ 2 ψ r ψ 2 r r 2 A2 s s s 2 s1 2 s1 2 s1
ψ 1 ψ 1 ψ 1
s3 s4 s3 s4 s3 s4 A3 A4 P = )|(),( dmp rrss P = )|(),( dmp rrss P = )|(),( dmp rrss 122 ∫ r 1 132 ∫ r 1 142 ∫ r 1 A A A Rensselaer Polytechnic2 Institute 3 Shivkumar Kalyanaraman4 87 : “shiv rpi” Upper bound based on minimum distance
P2 ik = Pr(),( zss is closer to k than i , when sss i is sent) ∞ 1 u 2 ⎛ d 2/ ⎞ = )exp( =− Qdu ⎜ ik ⎟ ∫ ⎜ ⎟ πN N0 N 2/ dik 0 ⎝ 0 ⎠
d −= ss kiik
1 M M ⎛ d 2/ ⎞ MP )( ≤ P ss −≤ )1(),( QM ⎜ min ⎟ E ∑∑ 2 ik ⎜ ⎟ M i=11k = N0 2/ ≠ik ⎝ ⎠
min = min dd ik Minimum distance in the signal space: ,ki ≠ki
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 88 : “shiv rpi” Example of upper bound on av. Symbol error prob. based on union bound
ψ 2 t)( si i s , iEE === 4,...,1
,ki = 2Ed s
≠ ki Es s2
= 2Ed d 2,1 min s d 3,2 s 3 s1 ψ 1 t)( − E s Es d 4,3 d 4,1
s4 − Es
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 89 : “shiv rpi” Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 90 : “shiv rpi” Eb/No figure of merit in digital communications SNR or S/N is the average signal power to the average noise power. SNR should be modified in terms of bit-energy in digital communication, because: Signals are transmitted within a symbol duration and hence, are energy signal (zero power). A metric at the bit-level facilitates comparison of different DCS transmitting different number of bits per symbol.
E ST S W Rb : Bit rate b b == : Bandwidth N0 /WN N Rb W Note: S/N = Eb/No x spectral efficiency Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 91 : “shiv rpi” Example of Symbol error prob. For PAM signals
Binary PAM s 2 s1 ψ 1 t)(
− Eb 0 Eb
4-ary PAM s s s 4 3 s 2 1
E E 0 E E ψ 1 t)( − 6 b − 2 b 2 b 6 b 5 5 5 5
ψ1 t)( 1 T
0 Tt
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 92 : “shiv rpi” Maximum Likelihood (ML) Detection: Vector Case
Nearest Neighbor Rule: By the isotropic property of the Gaussian noise, we expect the error
probability to be the same for both the transmit symbols uA, uB.
Error probability:
Project the received vector y along the
difference vector direction uA-uB is a “sufficient statistic”.
Noise outside these finite dimensions is irrelevant for detection. (rotational invariance of detection problem) ps: Vector norm is a natural extension of “magnitude” or length
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 93 : “shiv rpi” Extension to M-PAM (Multi-Level Modulation)
Note: h refers to the constellation shape/direction
MPAM:
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 94 : “shiv rpi” Complex Vector Space Detection
Error probability:
Note: Instead of vT, use v* for complex vectors (“transpose and conjugate”) for inner products…
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 95 : “shiv rpi” Complex Detection: Summary
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 96 : “shiv rpi” Detection Error => BER
If the bit error is i.i.d (discrete memoryless channel) over the sequence of bits, then you can model it as a binary symmetric channel (BSC) BER is modeled as a uniform probability f As BER (f) increases, the effects become increasingly intolerable f tends to increase rapidly with lower SNR: “waterfall” curve (Q-function)
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 97 : “shiv rpi” SNR vs BER: AWGN vs Rayleigh
Need diversity techniques to deal with Rayleigh (even 1-tap, flat-fading)!
Observe the “waterfall” like characteristic (essentially plotting the Q(x) function)! Telephone lines: SNR = 37dB, but low b/w (3.7kHz) Wireless: Low SNR = 5-10dB, higher bandwidth (upto 10 Mhz, MAN, and 20Mhz LAN) Optical fiber comm: High SNR, high bandwidth ! But cant process w/ complicated codes, signal processing etc Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 98 : “shiv rpi” Better performance through diversity
Diversity Ù the receiver is provided with multiple copies of the transmitted signal. The multiple signal copies should experience uncorrelated fading in the channel. In this case the probability that all signal copies fade simultaneously is reduced dramatically with respect to the probability that a single copy experiences a fade. As a rough rule: 1 DiversityDiversity ofof Pe is proportional to L LL:th:th order order γ 0
BERBER AverageAverage SNRSNR
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 99 : “shiv rpi” BER vs. SNR (diversity effect)
BER
()= Pe
Flat fading channel, Rayleigh fading, AWGN L = 1 channel (no fading)
SNR ()= γ 0 L = 4 L = 3 L = 2 We will explore this story later… slide set part II
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 100 : “shiv rpi” Modulation Techniques
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 101 : “shiv rpi” What is Modulation?
Encoding information in a manner suitable for transmission. Translate baseband source signal to bandpass signal Bandpass signal: “modulated signal”
How? Vary amplitude, phase or frequency of a carrier
Demodulation: extract baseband message from carrier
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 102 : “shiv rpi” Digital vs Analog Modulation
Cheaper, faster, more power efficient Higher data rates, power error correction, impairment resistance: Using coding, modulation, diversity Equalization, multicarrier techniques for ISI mitigation More efficient multiple access strategies, better security: CDMA, encryption etc
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 103 : “shiv rpi” Goals of Modulation Techniques
• High Bit Rate
• High Spectral Efficiency (max Bps/Hz)
• High Power Efficiency (min power to achieve a target BER)
• Low-Cost/Low-Power Implementation
• Robustness to Impairments
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 104 : “shiv rpi” Modulation: representation • Any modulated signal can be represented as
s(t) = A(t) cos [ωct+ φ(t)]
amplitude phase or frequency
s(t) = A(t) cos φ(t) cos ωct - A(t) sin φ(t) sin ωct in-phase quadrature
• Linear versus nonlinear modulation ⇒ impact on spectral efficiency Linear: Amplitude or phase Non-linear: frequency: spectral broadening • Constant envelope versus non-constant envelope ⇒ hardware implications with impact on power efficiency (=> reliability: i.e. target BER at lower SNRs)
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 105 : “shiv rpi” Complex Vector Spaces: Constellations
MPSK
Circular Square
Each signal is encoded (modulated) as a vector in a signal space
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 106 : “shiv rpi” Linear Modulation Techniques
s(t) = [ Σ an g (t-nT)]cos ωct-[ Σ bn g (t-nT)] sin ωct nn I(t), in-phase Q(t), quadrature
LINEAR MODULATIONS
Square M-ARY QUADRATURE M-ARY PHASE Circular Constellations AMPLITUDE MOD. SHIFT KEYING Constellations (M-QAM) (M-PSK)
M ≠ 4MM=4 ≠ 4 (4-QAM = 4-PSK)
CONVENTIONAL OFFSET DIFFERENTIAL 4-PSK 4-PSK 4-PSK (QPSK) (OQPSK) (DQPSK, π/4-DQPSK)
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 107 : “shiv rpi” M-PSK and M-QAM
M-PSK (Circular Constellations) M-QAM (Square Constellations)
bn bn 4-PSK 16-QAM 16-PSK 4-PSK
an an
Tradeoffs – Higher-order modulations (M large) are more spectrally efficient but less power efficient (i.e. BER higher). – M-QAM is more spectrally efficient than M-PSK but also more sensitive to system nonlinearities.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 108 : “shiv rpi” Bandwidth vs. Power Efficiency
MPSK:
MQAM:
MFSK:
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman Source: Rappaport book, chap 6 109 : “shiv rpi” MPAM & Symbol Mapping
Note: the average energy Binary PAM s 2 s 1 ψ t)( per-bit is constant 1 − Eb 0 Eb Gray coding used for 4-ary PAM s s s mapping bits to symbols 4 3 s 2 1
Why? Most likely error is to E E 0 E E ψ 1 t)( − 6 b − 2 b 2 b 6 b confuse with neighboring 5 5 5 5 symbol. Gray coding Make sure that the 00 01 11 10 neighboring symbol has only 4-ary PAM s s 3 s s 1-bit difference (hamming 4 2 1 ψ t)( distance = 1) E E 0 E E 1 − 6 b − 2 b 2 b 6 b 5 5 5 5
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 110 : “shiv rpi” MPAM: Details
Unequal energies/symbol:
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman Decision111 Regions : “shiv rpi” M-PSK (Circular Constellations)
bn MPSK: 4-PSK 16-PSK
an
Constellation points:
Equal energy in all signals:
01 01 00 Gray coding 11 00
10 Rensselaer Polytechnic Institute 10 11 Shivkumar Kalyanaraman 112 : “shiv rpi” MPSK: Decision Regions & Demod’ln
Z3 Z2 Z4 Z2
Z3 Z5 Z1 Z1
Z8 Z6 Z4 Z7 4PSK 8PSK 4PSK: 1 bit/complex dimension or 2 bits/symbol
m = 1 s (t) + n(t) i Z1: r > 0 m = 0 or 1 X g(Tb -t)
Z2: r ≤ 0 m = 0 cos(2πfct) Coherent Demodulator for BPSK.
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 113 : “shiv rpi” M-QAM (Square Constellations) MQAM: bn 16-QAM
4-PSK
an
Unequal symbol energies:
Z1 Z2 Z3 Z4 MQAM with square constellations of size L2 is equivalent to MPAM modulation with constellations of size L on each of the in-phase and quadrature signal components Z5 Z6 Z7 Z8
For square constellations it takes approximately Z Z Z Z 6 dB more power to send an additional 1 9 10 11 12 bit/dimension or 2 bits/symbol while maintaining the same minimum distance Z Z Z Z between constellation points 13 14 15 16
Hard to find a Gray code mapping where all adjacent symbols differ by a single bit 16QAM: Decision Regions Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 114 : “shiv rpi” Non-Coherent Modulation: DPSK
Information in MPSK, MQAM carried in signal phase.
Requires coherent demodulation: i.e. phase of the transmitted signal carrier φ0 must be matched to the phase of the receiver carrier φ More cost, susceptible to carrier phase drift. Harder to obtain in fading channels
Differential modulation: do not require phase reference. More general: modulation w/ memory: depends upon prior symbols transmitted. Use prev symbol as the a phase reference for current symbol Info bits encoded as the differential phase between current & previous symbol Less sensitive to carrier phase drift (f-domain) ; more sensitive to doppler effects: decorrelation of signal phase in time-domain
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 115 : “shiv rpi” Differential Modulation (Contd)
DPSK: Differential BPSK A 0 bit is encoded by no change in phase, whereas a 1 bit is encoded as a phase change of π. jθi If symbol over time [(k−1)Ts, kTs) has phase θ(k − 1) = e , θi = 0, π, then to encode a 0 bit over [kTs, (k + 1)Ts), the symbol would have phase: θ(k) = ejθi and… … to encode a 1 bit the symbol would have phase θ(k) = ej(θi+π).
DQPSK: gray coding:
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 116 : “shiv rpi” Quadrature Offset
Phase transitions of 180o can cause large amplitude transitions (through zero point). Abrupt phase transitions and large amplitude variations can be distorted by nonlinear amplifiers and filters
Avoided by offsetting the quadrature branch pulse g(t) by half a symbol period Usually abbreviated as O-MPSK, where the O indicates the offset QPSK modulation with quadrature offset is referred to as O-QPSK O-QPSK has the same spectral properties as QPSK for linear amplification,.. … but has higher spectral efficiency under nonlinear amplification, since the maximum phase transition of the signal is 90 degrees
Another technique to mitigate the amplitude fluctuations of a 180 degree phase shift used in the IS-54 standard for digital cellular is π/4-QPSK Maximum phase transition of 135 degrees, versus 90 degrees for offset QPSK and 180 degrees for QPSK Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 117 : “shiv rpi” Offset QPSK waveforms
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 118 : “shiv rpi” Frequency Shift Keying (FSK)
• Continuous Phase FSK (CPFSK) – digital data encoded in the frequency shift – typically implemented with frequency modulator to maintain continuous phase t s(t) = A cos [ωct+ 2 πkf ∫ d(τ) dτ] −∞ – nonlinear modulation but constant-envelope • Minimum Shift Keying (MSK) – minimum bandwidth, sidelobes large – can be implemented using I-Q receiver
• Gaussian Minimum Shift Keying (GMSK) – reduces sidelobes of MSK using a premodulation filter – used by RAM Mobile Data, CDPD, and HIPERLAN
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 119 : “shiv rpi” Minimum Shift Keying (MSK) spectra
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 120 : “shiv rpi” Spectral Characteristics 10 0 QPSK/DQPSK GMSK
-20
-40
-60 B3-dBTb = 0.16 0.25 -80 1.0
-100 Power Spectral Density (dB) (MSK)
-120 0 0.5 1.0 1.5 2.0 2.5 Normalized Frequency (f-fc)Tb Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 121 : “shiv rpi” Bit Error Probability (BER): AWGN 10-1 5
2 10-2 5 -3 For Pb = 10
2 DBPSK BPSK 6.5 dB -3 10 BPSK, QPSK QPSK 6.5 dB 5 DBPSK ~8 dB Pb DQPSK ~9 dB 2 DQPSK 10-4 5 • QPSK is more spectrally efficient than BPSK with the same performance. 2 • M-PSK, for M>4, is more spectrally efficient 10-5 but requires more SNR per bit. 5 • There is ~3 dB power penalty for differential 2 detection. 10-6 0 2 4 6 8 10 12 14 γb, SNR/bit, dB
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 122 : “shiv rpi” Bit Error Probability (BER): Fading Channel
1 5
2 10-1 5
2 10-2 5 DBPSK •P is inversely proportion to the average SNR per bit. Pb b 2 BPSK 10-3 • Transmission in a fading environment requires about -3 5 AWGN 18 dB more power for Pb = 10 .
2 10-4 5
2 10-5 0 5 101520253035 γb, SNR/bit, dB
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 123 : “shiv rpi” Bit Error Probability (BER): Doppler Effects
• Doppler causes an irreducible error floor when differential detection is used ⇒ decorrelation of reference signal.
100 QPSK DQPSK 10-1 Rayleigh Fading • The irreducible Pb depends on the data rate and the Doppler. -2 10 For fD = 80 Hz, data rate T P bfloor -3 -4 -4 P 10 10 kbps 10 s 3x10 b -5 -6 100 kbps 10 s 3x10 fDT=0.003 -6 -8 1 Mbps 10 s 3x10 10-4 No Fading 0.002 The implication is that Doppler is not an issue for high-speed 0.001 wireless data. -5 10 [M. D. Yacoub, Foundations of Mobile Radio Engineering , 0 CRC Press, 1993] 10-6 065040302010 0
γb, SNR/bit, dB
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 124 : “shiv rpi” Bit Error Probability (BER): Delay Spread
• ISI causes an irreducible error floor. 10-1 Coherent Detection +BPSK QPSK OQPSK Modulation x MSK • The rms delay spread imposes a limit on the x 10-2 maximum bit rate in a multipath environment. b x For example, for QPSK, x τ Maximum Bit Rate + Mobile (rural) 25 μsec 8 kbps + x Mobile (city) 2.5 μsec 80 kbps Microcells 500 nsec 400 kbps Irreducible P Irreducible + 10-3 Large Building 100 nsec 2 Mbps + [J. C.-I. Chuang, "The Effects of Time Delay Spread on Portable Radio x Communications Channels with Digital Modulation," IEEE JSAC, June 1987]
+
10-4 10-2 10-1 100 rms delay spread τ = symbol period T Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 125 : “shiv rpi” Summary of Modulation Issues
• Tradeoffs
– linear versus nonlinear modulation – constant envelope versus non-constant envelope – coherent versus differential detection – power efficiency versus spectral efficiency
• Limitations
– flat fading –doppler – delay spread
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 126 : “shiv rpi” Pulse Shaping
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 127 : “shiv rpi” Recall: Impact of AWGN only
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 128 : “shiv rpi” Impact of AWGN & Channel Distortion
c δ δ −−= Tttth )75.0(5.0)()(
Multi-tap, ISI channel
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 129 : “shiv rpi” ISI Effects: Band-limited Filtering of Channel
ISI due to filtering effect of the communications channel (e.g. wireless channels) Channels behave like band-limited filters
θc fj )( c = c )()( efHfH
Non-constant amplitude Non-linear phase
Amplitude distortion Phase distortion
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 130 : “shiv rpi” Inter-Symbol Interference (ISI)
ISI in the detection process due to the filtering effects of the system Overall equivalent system transfer function
= rct fHfHfHfH )()()()(
creates echoes and hence time dispersion causes ISI at sampling time ISI effect
kkk ++= ∑αi snsz i ≠ki
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 131 : “shiv rpi” Inter-symbol interference (ISI): Model
Baseband system model x1 x 2 z Channel Rx. filter k {}xk Tx filter tr )( {xˆk } t th )( c th )( r th )( Detector = kTt fH )( fH )( fH )( T x t c r 3 T tn )(
Equivalent model
x 1 x2 Equivalent system tz )( zk {}xk {xˆk } th )( Detector fH )( = kTt T x 3 T ˆ tn )( filtered noise
= rct fHfHfHfH )()()()(
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 132 : “shiv rpi” Nyquist bandwidth constraint
Nyquist bandwidth constraint (on equivalent system): The theoretical minimum required system bandwidth to detect Rs [symbols/s] without ISI is Rs/2 [Hz]. Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz] can support a maximum transmission rate of 2W=1/T=Rs [symbols/s] without ISI.
1 R R s W s ≥⇒≤= 2 [symbol/s/ Hz] T 22 W
Bandwidth efficiency, R/W [bits/s/Hz] : An important measure in DCs representing data throughput per hertz of bandwidth. Showing how efficiently the bandwidth resources are used by signaling techniques. Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 133 : “shiv rpi” Equiv System: Ideal Nyquist pulse (filter) Ideal Nyquist filter Ideal Nyquist pulse
fH )( = Ttth )/sinc()( T 1
− 2T −T 0 T 2T −1 0 1 f t 2T 2T 1 W = Rensselaer Polytechnic Institute 2T Shivkumar Kalyanaraman 134 : “shiv rpi” Nyquist pulses (filters)
Nyquist pulses (filters): Pulses (filters) which result in no ISI at the sampling time. Nyquist filter: Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even- symmetric frequency function Nyquist pulse: Its shape can be represented by a sinc(t/T) function multiply by another time function. Example of Nyquist filters: Raised-Cosine filter
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 135 : “shiv rpi” Pulse shaping to reduce ISI
Goals and trade-off in pulse-shaping Reduce ISI Efficient bandwidth utilization Robustness to timing error (small side lobes)
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 136 : “shiv rpi” Raised Cosine Filter: Nyquist Pulse Approximation
= RC fHfH |)(||)(| = RC thth )()(
1 r = 0 1
r = 5.0 0.5 0.5 r =1 r =1 r = 5.0 r = 0
−1 − 3 −1 0 1 3 1 − 3T − 2T −T 0 T 2T 3T T 4T 2T 2T 4T T
R )1( Baseband += rW )1( s Passband = + )1( RrW sSB 2 DSB s
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 137 : “shiv rpi” Raised Cosine Filter Raised-Cosine Filter A Nyquist pulse (No ISI at the sampling time)
⎧1 2||for 0 −< WWf ⎪ ⎪ 2 ⎡π −+ 2|| WWf 0 ⎤ fH )( = ⎨cos ⎢ ⎥ 0 ||2for <<− WfWW ⎪ ⎣ 4 −WW 0 ⎦ ⎩⎪0 ||for > Wf
π − 0 tWW ])(2cos[ = 0 0tWWth ))2(sinc(2)( 2 −− 0 tWW ])(4[1
−WW 0 Excess bandwidth: Roll-off factor r = −WW 0 W ≤ r ≤10 0
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 138 : “shiv rpi” Pulse Shaping and Equalization Principles
No ISI at the sampling time
RC = erct fHfHfHfHfH )()()()()(
Square-Root Raised Cosine (SRRC) filter and Equalizer
= fHfHfH )()()( RC rt Taking care of ISI caused by tr. filter r t == RC = SRRC fHfHfHfH )()()()( 1 e fH )( = Taking care of ISI c fH )( caused by channel
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 139 : “shiv rpi” Pulse Shaping & Orthogonal Bases
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 140 : “shiv rpi” Virtue of pulse shaping
PSD of a BPSK signal Rensselaer Polytechnic Institute Shivkumar Kalyanaraman Source: Rappaport book, chap 6 141 : “shiv rpi” Example of pulse shaping
Square-root Raised-Cosine (SRRC) pulse shaping Amp. [V]
Baseband tr. Waveform
Third pulse
t/T
First pulse Second pulse Data symbol
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 142 : “shiv rpi” Example of pulse shaping …
Raised Cosine pulse at the output of matched filter Amp. [V]
Baseband received waveform at the matched filter output (zero ISI)
t/T
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 143 : “shiv rpi” Eye pattern
Eye pattern:Display on an oscilloscope which sweeps the system response to a baseband signal at the rate 1/T (T symbol duration)
Distortion due to ISI Noise margin amplitude scale Sensitivity to timing error
Timing jitter
Rensselaer Polytechnic Institute timeShivkumar scale Kalyanaraman 144 : “shiv rpi” Example of eye pattern: Binary-PAM, SRRC pulse
Perfect channel (no noise and no ISI)
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 145 : “shiv rpi” Example of eye pattern: Binary-PAM, SRRC pulse …
AWGN (Eb/N0=20 dB) and no ISI
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 146 : “shiv rpi” Example of eye pattern: Binary-PAM, SRRC pulse …
AWGN (Eb/N0=10 dB) and no ISI
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 147 : “shiv rpi” Summary
Digital Basics Modulation & Detection, Performance, Bounds Modulation Schemes, Constellations Pulse Shaping
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 148 : “shiv rpi” Extra Slides
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 149 : “shiv rpi” Bandpass Modulation: I, Q Representation
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 150 : “shiv rpi” Analog: Frequency Modulation (FM) vs Amplitude Modulation (AM) FM: all information in the phase or frequency of carrier Non-linear or rapid improvement in reception quality beyond a minimum received signal threshold: “capture” effect. Better noise immunity & resistance to fading Tradeoff bandwidth (modulation index) for improved SNR: 6dB gain for 2x bandwidth Constant envelope signal: efficient (70%) class C power amps ok.
AM: linear dependence on quality & power of rcvd signal Spectrally efficient but susceptible to noise & fading Fading improvement using in-band pilot tones & adapt receiver gain to compensate Non-constant envelope: Power inefficient (30-40%) Class A or AB power amps needed: ½ the talk time as FM!
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 151 : “shiv rpi” Example Analog: Amplitude Modulation
Rensselaer Polytechnic Institute Shivkumar Kalyanaraman 152 : “shiv rpi”