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Point-to-Point Wireless Communication (I):Digital Basics, Modulation, Detection, Pulse Shaping

Point-to-Point Wireless Communication (I):Digital Basics, Modulation, Detection, Pulse Shaping 42
PointtoPoint Wireless Communication (I): Digital Basics, Modulation, Detection, Pulse Shaping Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 1The Basics Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 2Big Picture: Detection under AWGN Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 3Additive White Gaussian Noise (AWGN)  Thermal noise is described by a zeromean 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 nonzero lag w/Hz Power spectral Density (flat = “white”) Autocorrelation Function Probability density function (uncorrelated) (gaussian) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 4Effect 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 5Maximum Likelihood (ML) Detection: Scalar Case “likelihoods” Assuming both symbols equally likely: u is chosen if A A simple distance criterion LogLikelihood = Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 6AWGN Detection for Binary PAM p (z m ) z 2 p (z m ) z 1 s s 2 1  (t) 1 0  E E b b  s s / 2 1 2  P (m ) P (m ) Q e 1 e 2  N / 2 0   2E b  P P (2) Q B E  N 0  Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 7Bigger Picture General structure of a communication systems Noise Received Transmitted Received info. Info. signal signal SOURCE Transmitter Receiver Channel Source User Transmitter Source Channel Formatter Modulator encoder encoder Receiver Source Channel Formatter Demodulator decoder decoder Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 8Digital vs Analog Comm: Basics Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 9Digital 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 10Signal 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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 11Signal transmission (cont’d)  Ideal filters: Noncausal Lowpass Bandpass Highpass Duality = similar problems occur w/ rectangular pulses in time domain.  Realizable filters: RC filters Butterworth filter Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 12Bandwidth of signal  Baseband versus bandpass: Baseband Bandpass signal signal Local oscillator  Bandwidth dilemma: Bandlimited signals are not realizable Realizable signals have infinite bandwidth Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 13Bandwidth of signal: Approximations  Different definition of bandwidth: a) Halfpower bandwidth d) Fractional power containment bandwidth b) Noise equivalent bandwidth e) Bounded power spectral density c) Nulltonull bandwidth f) Absolute bandwidth (a) (b) (c) (d) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute (e)50dB : “shiv rpi” 14Formatting and transmission of baseband signal Digital info. Format Textual source info. Pulse Transmit Analog Encode Sample Quantize modulate info. Pulse Channel Bit stream waveforms Format Analog info. Lowpass Decode Demodulate/ filter Receive Detect Textual sink info. Digital info. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 15Sampling of Analog Signals Time domain Frequency domain x (t) x (t) x(t) s X ( f ) X ( f ) X ( f ) s x(t) X ( f ) X ( f ) x (t)  x (t) s X ( f ) s Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 16Aliasing effect Nyquist Rate LP filter Nyquist rate aliasing Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 17Undersampling Aliasing in Time Domain Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 18Nyquist Sampling Reconstruction: Time Domain Note: correct reconstruction does not draw straight lines between samples. Key: use sinc() pulses for reconstruction/interpolation Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 19Nyquist 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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 20Sampling theorem Sampling Analog Pulse amplitude signal modulated (PAM) signal process  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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 21Quantization  Amplitude quantizing: Mapping samples of a continuous amplitude waveform to a finite set of amplitudes. Out In Average quantization noise power Signal peak power Signal power to average quantization noise power Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 22 Quantized valuesEncoding (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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 23Quantization error  Quantizing error: The difference between the input and output of a quantizer ˆ e(t) x(t) x(t) Process of quantizing noise Qauntizer Model of quantizing noise y  q(x) AGC ˆ x(t) x(t) x(t) ˆ x(t) x e(t) + e(t) ˆ x(t) x(t) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 24Nonuniform 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 y C(x) ˆ x x(t) ˆ ˆ y(t) y(t) x(t) x ˆ y Compress Qauntize Expand Channel Transmitter Receiver Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 25Baseband 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. k  log M  Each transmit symbol represents bits of the PCM words. 2  PCM waveforms (line codes) are used for binary symbols (M=2).  Mary pulse modulation are used for nonbinary symbols (M2). Eg: Mary PAM.  For a given data rate, Mary PAM (M2) requires less bandwidth than binary PCM.  For a given average pulse power, binary PCM is easier to detect than M ary PAM (M2). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 26PAM example: Binary vs 8ary Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 27Example of Mary PAM Assuming real time Tx and equal energy per Tx data bit for binaryPAM and 4ary PAM: • 4ary: T=2T and Binary: T=T b b 2 2 A10B • Energy per symbol in binaryPAM: Binary PAM 4ary PAM (rectangular pulse) (rectangular pulse) 3B A. „11‟ B „1‟ T T „01‟ T T T „00‟ T B „10‟ „0‟ A. 3B Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 28Other PCM waveforms: Examples  PCM waveforms category:  Phase encoded  Nonreturntozero (NRZ)  Multilevel binary  Returntozero (RZ) 1 0 1 1 0 1 0 1 1 0 +V +V NRZL Manchester V V +V +V UnipolarRZ Miller 0 V +V +V BipolarRZ 0 0 Dicode NRZ V V 0 T 2T 3T 4T 5T 0 T 2T 3T 4T 5T Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 29PCM 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 30Summary: Baseband Formatting and transmission Digital info. Bit stream Pulse waveforms (Data bits) (baseband signals) Format Textual source info. Pulse Analog Encode Sample Quantize modulate info. Sampling at rate Encoding each q. value to f1/T bits l log L s s 2 (sampling time=Ts) (Data bit duration Tb=Ts/l) Mapping every m  l o g M data bits to a Quantizing each sampled 2 symbol out of M symbols and transmitting value to one of the a baseband waveform with duration T L levels in quantizer.  Information (data or bit) rate: R1/T bits/sec b b  Symbol rate : R1/T symbols/sec R mR Shivkumar Kalyanaraman Rensselaer Polytechnic Institute b : “shiv rpi” 31Receiver Structure Matched Filter Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 32Demodulation and detection g (t) s (t) m Pulse Bandpass i i Mary modulation i Format modulate modulate i1,,M channel transmitted symbol h (t) c estimated symbol n(t) Demod. Format Detect m ˆ sample z(T) r(t) i  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)  InterSymbol Interference (ISI)  Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared”. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 33Impact of AWGN Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 34Impact of AWGN Channel Distortion h (t) (t)0.5 (t0.75T) c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 35Receiver 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 36Receiver structure Step 1 – waveform to sample transformation Step 2 – decision making Demodulate Sample Detect z(T) ˆ m Threshold i r(t) Frequency Receiving Equalizing comparison downconversion filter filter Compensation for For bandpass signals channel induced ISI Baseband pulse Received waveform Sample Baseband pulse (possibly distorted) (test statistic) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 37Baseband 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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 38Steps 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 39Receiver filter to maximize the SNR  Model the received signal s (t) r(t) h (t) i r(t) s (t)h (t) n(t) c i c n(t) AWGN  Simplify the model (ideal channel assumption):  Received signal in AWGN Ideal channels r(t) s (t) r(t) s (t) n(t) i i h (t) (t) c n(t) AWGN Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 40Matched Filter Receiver  Problem:  Design the receiver filter such that the SNR is h(t) maximized at the sampling time when s (t), i1,...,M i is transmitted.  Solution:  The optimum filter, is the Matched filter, given by h(t) h (t) s (Tt) opt i H( f ) H ( f ) S ( f )exp( j2fT ) opt i which is the timereversed and delayed version of the conjugate of the transmitted signal h(t) h (t) s (t) i opt 0 T t 0 T t Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 41Correlator Receiver  The matched filter output at the sampling time, can be realized as the correlator output.  Matched filtering, i.e. convolution with s (T) simplifies to i integration w/ s (), i.e. correlation or inner product i z(T) h (T)r(T) opt T  r( )s ( )d  r(t), s(t) 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 42Irrelevance 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). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 43Aside: 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”,  ) 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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 44Aside: Autocorrelation Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 45Aside: CrossCorrelation 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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 46• 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Source: Bores Signal Processing 47Matched 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 signaltonoise ratio it can be shown that it also minimizes the BER: it is a simple projection operation Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 48Example of matched filter (real signals) y(t) s (t)h (t) i opt 2 s (t) h (t) A i opt A A T T 0 2T T t T t T t y(t) s (t)h (t) i opt 2 s (t) h (t) A i opt A A T T 0 T/2 3T/2 2T T/2 T t T/2 T T t T t 2 A  A A  2 T T Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 49Properties 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 Z( f ) S( f ) exp( j2fT ) 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. z(t) R (tT) z(T) R (0) E s s 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  s max  N N / 2  T 0 4. Two matching conditions in the matchedfiltering operation:  spectral phase matching that gives the desired output peak at time T.  spectral amplitude matching that gives optimum SNR to the peak value. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 50Implementation of matched filter receiver Bank of M matched filters z (T) 1 s (Tt) 1 z  1 Matched filter output: r(t) z Observation   z  vector   z M  s (Tt) M z (T) M  i1,...,M z r(t)s (Tt) i i z (z (T), z (T),..., z (T)) (z , z ,..., z ) 1 2 M 1 2 M Note: we are projecting along the basis directions of the signal space Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 51Implementation of correlator receiver Bank of M correlators  s (t) 1 T z (T) 1 z   Correlators output: 1 0 r(t) z Observation   z   vector s (t) M  T  z M   z (T) 0 M z (z (T), z (T),..., z (T)) (z , z ,..., z ) 1 2 M 1 2 M T z r(t)s (t)dt i1,...,M i i  0 Note: In previous slide we “filter” i.e. convolute in the boxes shown. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 52Implementation example of matched filter receivers s (t) 1 Bank of 2 matched filters A T z (T) A 0 1 T t T z  1 r(t) z   z 0 T  s (t) 2  z 2  T 0 z (T) 2 0 T t  A A T T Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 53Matched Filter: Frequency domain View Simple Bandpass Filter: excludes noise, but misses some signal power Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 54Matched Filter: Frequency Domain View (Contd) MultiBandpass Filter: includes more signal power, but adds more noise also Matched Filter: includes more signal power, weighted according to size = maximal noise rejection Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 55Maximal Ratio Combining (MRC) viewpoint  Generalization of this fdomain picture, for combining multitap signal Weight each branch SNR: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 56Examples of matched filter output for bandpass modulation schemes Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 57Signal Space Concepts Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 58Signal space: Overview  What is a signal space  Vector representations of signals in an Ndimensional 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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 59Schematic example of a signal space  (t) 2 s (a ,a ) 1 11 12  (t) 1 z (z , z ) 1 2 s (a ,a ) 3 31 32 s (a ,a ) 2 21 22 s (t) a (t) a (t) s (a ,a ) 1 11 1 12 2 1 11 12 Transmitted signal s (t) a (t) a (t) s (a ,a ) 2 21 1 22 2 2 21 22 alternatives s (t) a (t) a (t) s (a ,a ) 3 31 1 32 2 3 31 32 Received signal at z(t) z (t) z (t) z (z , z ) 1 1 2 2 1 2 matched filter output Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 60Signal space  To form a signal space, first we need to know the inner product between two signals (functions):  Inner (scalar) product:   x(t), y(t)  x(t)y (t)dt   = crosscorrelation between x(t) and y(t)  Properties of inner product:  ax(t), y(t) a x(t), y(t)  x(t),ay(t)  a x(t), y(t)  x(t) y(t), z(t) x(t), z(t) y(t), z(t) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 61Signal space …  The distance in signal space is measure by calculating the norm.  What is norm  Norm of a signal:  2 x(t) x(t), x(t) x(t) dt E x   = “length” of x(t) ax(t) a x(t)  Norm between two signals: d x(t) y(t) x,y  We refer to the norm between two signals as the Euclidean distance between two signals. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 62Example of distances in signal space  (t) 2 s (a ,a ) 1 11 12 E d 1 s ,z 1  (t) 1 E 3 z (z , z ) 1 2 d E s ,z d 3 2 s ,z 2 s (a ,a ) 3 31 32 s (a ,a ) 2 21 22 The Euclidean distance between signals z(t) and s(t): 2 2 d s (t) z(t) (a z ) (a z ) s ,z i i1 1 i2 2 i i1,2,3 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 63Orthogonal signal space  Ndimensional orthogonal signal space is characterized by N N   (t) linearly independent functions called basis functions. j j1 The basis functions must satisfy the orthogonality condition T 0 t T  (t), (t)  (t) (t)dt K i j i j i ji  j,i1,..., N 0 where 1 i j    ij 0 i j   If all , the signal space is orthonormal. K1 i  Constructing Orthonormal basis from nonorthonormal set of vectors:  GramSchmidt procedure Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 64Example of an orthonormal bases  Example: 2dimensional orthonormal signal space  2  (t) cos(2t /T ) 0 t T  (t)  1 2  T  2   (t) sin(2t /T ) 0 t T 2  T   (t) 1 T 0  (t), (t)  (t) (t)dt 0 1 2 1 2  0  (t) (t)1 1 2  Example: 1dimensional orthonornal signal space  (t) 1 1  (t)1 1  (t) T 1 0 0 T t Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 65Sine/Cosine Bases: Note  Approximately orthonormal  These are the inphase quadraturephase dimensions of complex baseband equivalent representations. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 66Example: BPSK  Note: two symbols, but only one dimension in BPSK. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 67Signal space … M  Any arbitrary finite set of waveforms s (t) i i1 where each member of the set is of duration T, can be expressed as a linear combination of N orthogonal waveforms N where .  (t) N M j j1 N i1,...,M s (t) a (t) i ij j j1 N M where T 1 1 j1,..., N a s (t), (t)  s (t) (t)dt 0 t T ij i j i j  i1,...,M K K j j 0 N 2 s (a ,a ,...,a ) E K a i i1 i2 iN i j ij j1 Vector representation of waveform Waveform energy Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 68Signal space … N s (t) a (t) s (a ,a ,...,a ) i ij j i i1 i2 iN j1 Waveform to vector conversion Vector to waveform conversion  (t) 1  (t) 1 a T i1 a i1 a  a   i1 i1 0 s s (t) m i s (t)   i  s   m  (t)  (t) N N  T  a  a iN  iN  a  a iN 0 iN Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 69Example of projecting signals to an orthonormal signal space  (t) 2 s (a ,a ) 1 11 12  (t) 1 s (a ,a ) 3 31 32 s (a ,a ) 2 21 22 s (t) a (t) a (t) s (a ,a ) 1 11 1 12 2 1 11 12 Transmitted signal s (t) a (t) a (t) s (a ,a ) alternatives 2 21 1 22 2 2 21 22 s (t) a (t) a (t) s (a ,a ) 3 31 1 32 2 3 31 32 T a s (t) (t)dt j1,..., N Shivkumar Kalyanaraman i1,...,M Rensselaer Polytechnic Institute ij i j 0 t T  0 : “shiv rpi” 70Matched filter receiver (revisited) (note: we match to the basis directions) Bank of N matched filters z 1  Observation  (Tt) 1 z  vector 1 r(t) z   z   z N  (Tt) N z N N s (t) a (t) i ij j i1,...,M j1 z (z , z ,..., z ) N M 1 2 N j1,..., N z r(t) (Tt) j j Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 71Correlator receiver (revisited) Bank of N correlators  (t) 1 T z 1 r   1 0 r(t) z Observation   z  vector  (t) N  T  r N  z  N 0 N s (t) a (t) i1,...,M i ij j j1 z (z , z ,..., z ) 1 2 N N M T j1,..., N z r(t) (t)dt j j  Shivkumar Kalyanaraman 0 Rensselaer Polytechnic Institute : “shiv rpi” 72Example of matched filter receivers using basic functions s (t) s (t)  (t) 1 2 1 A 1 T T T t 0 0 T t 0 A T t T 1 matched filter  (t) 1 1 r(t) z z T 1  z z 1 0 T t  Number of matched filters (or correlators) is reduced by 1 compared to using matched filters (correlators) to the transmitted signal Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 73White noise in Orthonormal Signal Space  AWGN n(t) can be expressed as ˆ n(t) n(t) n(t) Noise outside on the signal space Noise projected on the signal space (irrelevant) (colored):impacts the detection process. N ˆ n(t) n (t)  j j ˆ n(t) Vector representation of j1 n (n ,n ,...,n ) n  n(t), (t) 1 2 N j1,..., N j j N n independent zeromean j j1,..., N  n(t), (t)  0 j1 j Gaussain random variables with var(n ) N / 2 variance j 0 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 74Detection: Maximum Likelihood Performance Bounds Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 75Detection of signal in AWGN  Detection problem:  Given the observation vector , perform a mapping from z z ˆ m m to an estimate of the transmitted symbol, , such that i the average probability of error in the decision is minimized. n s z i m ˆ m Modulator Decision rule i Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 76Statistics of the observation Vector z s n  AWGN channel model: i  Signal vector s (a ,a ,...,a ) is deterministic. i i1 i2 iN n (n ,n ,...,n )  Elements of noise vector are i.i.d Gaussian 1 2 N random variables with zeromean and variance . The N / 2 0 noise vector pdf is 2  n 1  p (n) exp n N / 2  N N 0 0   The elements of observed vector are z (z , z ,..., z ) 1 2 N independent Gaussian random variables. Its pdf is 2  z s 1 i  p (z s ) exp z i N / 2  N N 0 0  Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 77Detection  Optimum decision rule (maximum a posteriori probability): ˆ Set m m if i Pr(m sent z) Pr(m sent z), for all k i i k where k1,..., M.  Applying Bayes‟ rule gives: ˆ Set m m if i p (z m ) z k p , is maximum for all k i k p (z) z Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 78Detection …  Partition the signal space into M decision regions, such that Z ,..., Z 1 M Vector z lies inside region Z if i p (z m ) z k ln p , is maximum for all k i. k p (z) z That means ˆ m m i Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 79Detection (ML rule)  For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to: ˆ Set m m if i p (z m ), is maximum for all k i z k or equivalently: ˆ Set m m if i ln p (z m ), is maximum for all k i z k which is known as maximum likelihood. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 80Detection (ML)… Z ,..., Z  Partition the signal space into M decision regions, 1 M  Restate the maximum likelihood decision rule as follows: Vector z lies inside region Z if i ln p (z m ), is maximum for all k i z k That means ˆ m m i Vector z lies inside region Z if i z s , is minimum for all k i k Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 81Schematic example of ML decision regions  (t) 2 Z 2 s 2 Z 1 s s 3 1  (t) 1 Z 3 s 4 Z 4 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 82Probability of symbol error  Erroneous decision: For the transmitted symbol s or equivalently signal i vector , an error in decision occurs if the observation vector does not m z i fall inside region . Z i  Probability of erroneous decision for a transmitted symbol ˆ Pr(m m ) Pr(m sent)Pr(z does not lie inside Z m sent) i i i i  Probability of correct decision for a transmitted symbol ˆ Pr(m m ) Pr(m sent)Pr(z lies inside Z m sent) i i i i P (m ) Pr(z lies inside Z m sent) p (z m )dz c i i i z i  Z i P (m )1 P (m ) e i c i Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 83Example for binary PAM p (z m ) z 2 p (z m ) z 1 s s 2 1  (t) 1 0  E E b b  s s / 2 1 2  P (m ) P (m ) Q e 1 e 2  N / 2 0   2E b  P P (2) Q B E  N 0  Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 84Average prob. of symbol error …  Average probability of symbol error : M ˆ P (M ) Pr(m m ) E i i1  For equally probable symbols: M M 1 1 P (M ) P (m )1 P (m )  E e i c i M M i1 i1 M 1 1 p (z m )dz  z i  M i1 Z i Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 85Union 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 M M 1 P (m ) P (s ,s ) P (M ) P (s ,s )  e i 2 k i E 2 k i M k1 i1 k1 ki ki Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 86Example of union bound  2 Z Z r 2 1 P (m ) p (r m )dr e 1 r 1 s  2 s 1 ZZZ 2 3 4  1 Union bound: s s 3 4 4 Z Z 3 4 P (m ) P (s ,s )  e 1 2 k 1 k2   2  2 r r r 2 A 2 s s s 2 s 2 s 2 s 1 1 1  1 1 1 s s s s s s 3 4 3 4 3 4 A A 4 3 P (s ,s ) p (r m )dr P (s ,s ) p (r m )dr P (s ,s ) p (r m )dr 2 2 1 r 1 2 3 1 r 1 2 4 1 r 1   A A A 2 3 4 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 87Upper bound based on minimum distance P (s ,s ) Pr(z is closer to s than s , when s is sent) 2 k i k i i  2  1 u d / 2 ik   exp( )duQ   N N N / 2 0 d 0 0  ik d ss ik i k M M  1 d / 2 min  P (M ) P (s ,s ) (M1)Q  E 2 k i  M N / 2 i1 k1 0  ki d min d min ik Minimum distance in the signal space: i,k ik Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 88Example of upper bound on av. Symbol error prob. based on union bound  (t) 2 s E E , i1,...,4 i i s d 2E i,k s i k s E s 2 d d 2E 1,2 d min s 2,3 s s 3 1  (t) 1  E E s s d d 3,4 1,4 s 4  E s Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 89Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 90Eb/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 bitenergy in digital communication, because:  Signals are transmitted within a symbol duration and hence, are energy signal (zero power).  A metric at the bitlevel facilitates comparison of different DCS transmitting different number of bits per symbol. R : Bit rate E ST S W b b b  : Bandwidth W N N /W N R 0 b Note: S/N = Eb/No x spectral efficiency Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 91Example of Symbol error prob. For PAM signals Binary PAM s s 2 1  (t) 1 0 E  E b b 4ary PAM s s s s 4 3 1 2  (t) 0 E E E E 1 b b b b  6 6  2 2 5 5 5 5  (t) 1 1 T 0 T t Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 92Maximum 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 u , u . A B Error probability: Project the received vector y along the difference vector direction u u is a A B “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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 93Extension to MPAM (MultiLevel Modulation)  Note: h refers to the constellation shape/direction MPAM: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 94Complex Vector Space Detection Error probability: T  Note: Instead of v , use v for complex vectors (“transpose and conjugate”) for inner products… Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 95Complex Detection: Summary Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 96Detection 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 (Qfunction) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 97SNR vs BER: AWGN vs Rayleigh Need diversity techniques to deal with Rayleigh (even 1tap, flatfading)  Observe the “waterfall” like characteristic (essentially plotting the Q(x) function)  Telephone lines: SNR = 37dB, but low b/w (3.7kHz)  Wireless: Low SNR = 510dB, 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 98Better performance through diversity Diversity  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 Diversity of P is proportional to e L L:th order  0 BER Average SNR Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 99BER vs. SNR (diversity effect) BER () P e Flat fading channel, Rayleigh fading, AWGN L = 1 channel (no fading) () SNR 0 L = 4 L = 3 L = 2 We will explore this story later… slide set part II Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 100Modulation Techniques Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 101What 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 102Digital 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 103Goals of Modulation Techniques • High Bit Rate • High Spectral Efficiency (max Bps/Hz) • High Power Efficiency (min power to achieve a target BER) • LowCost/LowPower Implementation • Robustness to Impairments Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 104Modulation: representation • Any modulated signal can be represented as s(t) = A(t) cos w t + f(t) c amplitude phase or frequency s(t) A(t) sin f(t) sin w t = A(t) cos f(t) cos w t c c quadrature inphase • Linear versus nonlinear modulation impact on spectral efficiency Linear: Amplitude or phase Nonlinear: frequency: spectral broadening • Constant envelope versus nonconstant envelope hardware implications with impact on power efficiency (= reliability: i.e. target BER at lower SNRs) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 105Complex Vector Spaces: Constellations MPSK Circular Square  Each signal is encoded (modulated) as a vector in a signal space Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 106Linear Modulation Techniques s(t) = S a g (tnT)cos w t S b g (tnT) sin w t n c n c n n I (t), inphase Q(t), quadrature LINEAR MODULATIONS MARY QUADRATURE MARY PHASE Square Circular AMPLITUDE MOD. SHIFT KEYING Constellations Constellations (MQAM) (MPSK) M=4 M 4 M 4 (4QAM = 4PSK) CONVENTIONAL OFFSET DIFFERENTIAL 4PSK 4PSK 4PSK (QPSK) (OQPSK) (DQPSK, /4DQPSK) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 107MPSK and MQAM MQAM (Square Constellations) MPSK (Circular Constellations) b b n n 4PSK 16QAM 16PSK 4PSK a an n Tradeoffs – Higherorder modulations (M large) are more spectrally efficient but less power efficient (i.e. BER higher). – MQAM is more spectrally efficient than MPSK but also more sensitive to system nonlinearities. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 108Bandwidth vs. Power Efficiency MPSK: MQAM: MFSK: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: Rappaport book, chap 6 : “shiv rpi” 109MPAM Symbol Mapping Binary PAM  Note: the average energy s s 2 1  (t) 1 perbit is constant 0 E  E b b  Gray coding used for 4ary PAM s s s mapping bits to symbols s 4 3 1 2  (t) 0  Why Most likely error is to E E E E 1 b b b b  6 6  2 2 5 5 5 5 confuse with neighboring symbol. Gray coding  Make sure that the 01 11 10 00 neighboring symbol has only 4ary PAM s s s s 4 3 1 2 1bit difference (hamming  (t) 0 distance = 1) E E E E 1 b b b b  6 6  2 2 5 5 5 5 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 110MPAM: Details Unequal energies/symbol: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Decision Regions : “shiv rpi” 111MPSK (Circular Constellations) b n MPSK: 4PSK 16PSK an  Constellation points:  Equal energy in all signals: 01 01 00  Gray coding 00 11 10 11 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 10 : “shiv rpi” 112MPSK: Decision Regions Demod’ln Z 3 Z Z 2 4 Z 2 Z Z 3 5 Z Z 1 1 Z 8 Z 6 Z Z 4 7 4PSK 8PSK 4PSK: 1 bit/complex dimension or 2 bits/symbol m = 1 s (t) + n(t) Z : r 0 i 1 m = 0 or 1 X g(T t) b Z : r ≤ 0 2 m = 0 cos(2πf t) c Coherent Demodulator for BPSK. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 113MQAM (Square Constellations) b n MQAM: 16QAM 4PSK a n  Unequal symbol energies: Z Z Z Z 1 2 3 4 2  MQAM with square constellations of size L is equivalent to MPAM modulation with constellations of size L on each of the inphase Z Z Z Z 5 6 7 8 and quadrature signal components  For square constellations it takes approximately Z Z Z Z 9 10 11 12 6 dB more power to send an additional 1 bit/dimension or 2 bits/symbol while maintaining the same minimum distance Z Z Z Z 13 14 15 16 between constellation points  Hard to find a Gray code mapping where all 16QAM: Decision Regions adjacent symbols differ by a single bit Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 114NonCoherent Modulation: DPSK  Information in MPSK, MQAM carried in signal phase.  Requires coherent demodulation: i.e. phase of the transmitted signal carrier φ must 0 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 (fdomain) ; more sensitive to doppler effects: decorrelation of signal phase in timedomain Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 115Differential 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)T , kT ) has phase θ(k − 1) = e , θ = 0, π, s s i  then to encode a 0 bit over kTs, (k + 1)Ts), the symbol would have jθi  phase: θ(k) = e and…  … to encode a 1 bit the symbol would have j(θi+π)  phase θ(k) = e .  DQPSK: gray coding: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 116Quadrature Offset o  Phase transitions of 180 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 OMPSK, where the O indicates the offset  QPSK modulation with quadrature offset is referred to as OQPSK  OQPSK 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 IS54 standard for digital cellular is π/4QPSK  Maximum phase transition of 135 degrees, versus 90 degrees for offset QPSK and 180 degrees for QPSK Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 117Offset QPSK waveforms Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 118Frequency 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 w t + 2 k d() d c f  – nonlinear modulation but constantenvelope • Minimum Shift Keying (MSK) – minimum bandwidth, sidelobes large – can be implemented using IQ receiver • Gaussian Minimum Shift Keying (GMSK) – reduces sidelobes of MSK using a premodulation filter – used by RAM Mobile Data, CDPD, and HIPERLAN Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 119Minimum Shift Keying (MSK) spectra Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 120Spectral Characteristics 10 QPSK/DQPSK 0 GMSK 20 40 60 B T = 0.16 3dB b 0.25 80 1.0 100 (MSK) 120 0 0.5 1.0 1.5 2.0 2.5 Normalized Frequency (ff )T c b Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 121 Power Spectral Density (dB)Bit Error Probability (BER): AWGN 1 10 5 2 2 10 3 5 For P = 10 b 2 BPSK 6.5 dB DBPSK 3 QPSK 6.5 dB 10 BPSK, QPSK DBPSK 8 dB 5 P DQPSK 9 dB b 2 DQPSK 4 10 • QPSK is more spectrally efficient than BPSK 5 with the same performance. 2 • MPSK, for M4, is more spectrally efficient 5 10 but requires more SNR per bit. 5 • There is 3 dB power penalty for differential 2 detection. 6 10 0 2 4 6 8 10 12 14  , SNR/bit, dB b Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 122Bit Error Probability (BER): Fading Channel 1 5 2 1 10 5 2 2 10 DBPSK 5 • P is inversely proportion to the average SNR per bit. P b b 2 BPSK 3 10 • Transmission in a fading environment requires about 3 18 dB more power for P = 10 . 5 AWGN b 2 4 10 5 2 5 10 0 5 10 15 20 25 30 35  , SNR/bit, dB b Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 123Bit Error Probability (BER): Doppler Effects • Doppler causes an irreducible error floor when differential detection is used decorrelation of reference signal. 0 10 QPSK DQPSK 1 10 Rayleigh Fading • The irreducible P depends on the data rate and the Doppler. b 2 For f = 80 Hz, 10 D data rate T P b floor 3 4 4 10 10 kbps 10 3x10 s P b 5 6 100 kbps 10 s 3x10 f T=0.003 D 6 8 1 Mbps 10 s 3x10 4 No Fading 10 0.002 The implication is that Doppler is not an issue for highspeed 0.001 wireless data. 5 10 M. D. Yacoub, Foundations of Mobile Radio Engineering , CRC Press, 1993 0 6 10 0 10 20 30 40 50 60  , SNR/bit, dB b Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 124Bit Error Probability (BER): Delay Spread • ISI causes an irreducible error floor. 1 10 Coherent Detection +BPSK QPSK Modulation OQPSK x MSK • The rms delay spread imposes a limit on the x 2 maximum bit rate 10 in a multipath environment. x For example, for QPSK,  Maximum Bit Rate x + Mobile (rural) 25 msec 8 kbps + Mobile (city) 2.5 msec 80 kbps x Microcells 500 nsec 400 kbps + Large Building 100 nsec 2 Mbps 3 10 + J. C.I. Chuang, "The Effects of Time Delay Spread on Portable Radio x Communications Channels with Digital Modulation," IEEE JSAC, June 1987 + 4 10 2 1 0 10 10 10 rms delay spread  = symbol period T Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 125 Irreducible P bSummary of Modulation Issues • Tradeoffs – linear versus nonlinear modulation – constant envelope versus nonconstant envelope – coherent versus differential detection – power efficiency versus spectral efficiency • Limitations – flat fading – doppler – delay spread Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 126Pulse Shaping Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 127Recall: Impact of AWGN only Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 128Impact of AWGN Channel Distortion h (t) (t)0.5 (t0.75T) c Multitap, ISI channel Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 129ISI Effects: Bandlimited Filtering of Channel  ISI due to filtering effect of the communications channel (e.g. wireless channels)  Channels behave like bandlimited filters j ( f ) c H ( f ) H ( f ) e c c Nonconstant amplitude Nonlinear phase Amplitude distortion Phase distortion Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 130InterSymbol Interference (ISI)  ISI in the detection process due to the filtering effects of the system  Overall equivalent system transfer function H( f ) H ( f )H ( f )H ( f ) t c r  creates echoes and hence time dispersion  causes ISI at sampling time ISI effect z s n s  k k k i i ik Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 131Intersymbol interference (ISI): Model  Baseband system model x x 1 2 z k Channel Rx. filter x Tx filter ˆ r(t)x k k h (t) h (t) h (t) t c r Detector t kT H ( f ) H ( f ) H ( f ) t c r T x T 3 n(t)  Equivalent model x x 1 2 Equivalent system z k z(t) x ˆ x k k h(t) Detector t kT H ( f ) T x T 3 ˆ n(t) filtered noise H( f ) H ( f )H ( f )H ( f ) t c r Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 132Nyquist bandwidth constraint  Nyquist bandwidth constraint (on equivalent system):  The theoretical minimum required system bandwidth to detect R symbols/s without ISI is R /2 Hz. s s  Equivalently, a system with bandwidth W=1/2T=R /2 Hz s can support a maximum transmission rate of 2W=1/T=Rs symbols/s without ISI. 1 R R s s W 2 symbol/s/Hz 2T 2 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. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 133Equiv System: Ideal Nyquist pulse (filter) Ideal Nyquist filter Ideal Nyquist pulse H( f ) h(t) sinc(t /T) T 1 0  2TT T 2T 0 f t 1 1 2T 2T 1 W 2T Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 134Nyquist 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: RaisedCosine filter Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 135Pulse shaping to reduce ISI  Goals and tradeoff in pulseshaping  Reduce ISI  Efficient bandwidth utilization  Robustness to timing error (small side lobes) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 136Raised Cosine Filter: Nyquist Pulse Approximation H( f )  H ( f ) h(t) h (t) RC RC 1 1 r 0 r 0.5 r1 0.5 0.5 r1 r 0.5 r 0  2T T 3T 1 313T T 2T 0 0 1 3 1 T 4T 2T 2T 4T T R s Baseband W (1 r) Passband W (1 r)R sSB DSB s 2 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 137Raised Cosine Filter  RaisedCosine Filter  A Nyquist pulse (No ISI at the sampling time) 1 for f  2WW  0    f W 2W  2 0 H ( f ) cos for 2WW f  W   0 4 WW 0   0 for f  W  cos2 (WW )t 0 h(t) 2W (sinc(2W t)) 0 0 2 14(WW )t 0 WW 0 r Rolloff factor Excess bandwidth: WW 0 W 0 0 r1 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 138Pulse Shaping and Equalization Principles No ISI at the sampling time H ( f ) H ( f )H ( f )H ( f )H ( f ) RC t c r e  SquareRoot Raised Cosine (SRRC) filter and Equalizer H ( f ) H ( f )H ( f ) RC t r Taking care of ISI caused by tr. filter H ( f ) H ( f ) H ( f ) H ( f ) r t RC SRRC 1 Taking care of ISI H ( f ) e H ( f ) caused by channel c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 139Pulse Shaping Orthogonal Bases Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 140Virtue of pulse shaping PSD of a BPSK signal Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: Rappaport book, chap 6 : “shiv rpi” 141Example of pulse shaping  Squareroot RaisedCosine (SRRC) pulse shaping Amp. V Baseband tr. Waveform Third pulse t/T First pulse Second pulse Data symbol Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 142Example 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 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 143Eye 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 Sensitivity to timing error Timing jitter time scale Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 144 amplitude scaleExample of eye pattern: BinaryPAM, SRRC pulse  Perfect channel (no noise and no ISI) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 145Example of eye pattern: BinaryPAM, SRRC pulse …  AWGN (Eb/N0=20 dB) and no ISI Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 146Example of eye pattern: BinaryPAM, SRRC pulse …  AWGN (Eb/N0=10 dB) and no ISI Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 147Summary  Digital Basics  Modulation Detection, Performance, Bounds  Modulation Schemes, Constellations  Pulse Shaping Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 148Extra Slides Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 149Bandpass Modulation: I, Q Representation Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 150Analog: Frequency Modulation (FM) vs Amplitude Modulation (AM)  FM: all information in the phase or frequency of carrier  Nonlinear 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 inband pilot tones adapt receiver gain to compensate  Nonconstant envelope: Power inefficient (3040) Class A or AB power amps needed: ½ the talk time as FM Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 151Example Analog: Amplitude Modulation Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 152
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