Chapter 2: Signal Representation

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1 Chapter 2: Signal Representation Aveek Dutta Assistant Professor Department of Electrical and Computer Engineering University at Albany Spring 2018 Images and equations adopted from: Digital Communications - John G. Proakis and Masoud Salehi 5/e. Copyright by The McGraw Hill Companies Fundamentals of Communication Systems - Michael P. Fitz,.Copyright by The McGraw Hill Companies

2 Recap of Lecture 1 Signals are time domain functions Waveforms are distorted by channel noise, which is not deterministic Use statistical models to approximate noise The ratio of Signal Power to Noise Power is called SNR (Signal to Noise Ratio). Unit is db Goal is to design systems that maximize SNR Receivers are inverse operations of the transmitter Converted to binary streams (Sampling and Quantization) Add redundancy (Coding Theory) Map bits to Waveform (Modulation - Analog or Digital). Sometimes called upconversion Higher frequency waveform (Carrier) carry lower frequency information signals Pulse shaping, filtering, etc. - Mostly to conform to mandated spectrum Use amplifiers and antenna to convert electrical signals to EM waves Signals occupy bandwidth (Hz) - Thanks to Joseph Fourier. Antenna converts EM to electrical signals Condition incoming signal (Equalization) Demod, Decode, Interpolate to reproduce the transmitted signal Key metric is BER (Bit Error Rate) Higher the SNR, lower the BER while Maximizing Bits/sec/Hz (Spectral Efficiency)

3 Background Channel - Type 1 Channel - Type 2 Channel - Type 3 Signals can be: Deterministic or Random Periodic or Aperiodic Complex or Real Continuous or Discrete Time Common Signals

4 Fourier Transform Image from: Read These - even if you think you know FT Fourier Transform Pair

5 2.1 - Bandpass and Lowpass Signals A bandpass signal, xc(t), is a signal whose one-sided energy spectrum is both: centered at a nonzero frequency, f C does not extend in frequency to zero (DC) A real-valued bandpass signal, x(t), has Hermitian Symmetry X(-f) = X*(f), from which we conclude that X(-f) = X(f) and X*(f) = - X(f). In other words, for real x(t), the magnitude of X(f) is even and phase is odd A lowpass or baseband signal has its spectrum located around the zero frequency (DC) Also define +ve and -ve spectrum Therefore, X(f) = X+(f) + X_(f) For a real signal x(t), since X(f) is Hermitian, we have X_(f) = X*+(-f).

6 Low pass equivalent of a bandpass signal Say, there exists a signal x+(t), corresponding to the signal x(t) with just the +ve spectrum X+(f) Only consider the +ve frequencies Hilbert Transform of x(t) xl(t) is the lowpass equivalent of the bandpass signal, whose spectrum is defined by 2X+(f+f0) Modulation Theorem of FT

7 ...contd Modulation Theorem of FT Definition: The real and imaginary parts of xl(t) are called the in-phase component and the quadrature component of x(t), respectively, and are denoted by Xi(t) and xq(t). Solving for x(t) and x^(t)

8 Mod - Demod Modulator Demodulator Equivalent Structure

9 Visualization of Baseband Signal

10 Example x x

11 Energy of signals The energy of signal x(t) is given by Energy of the one sided spectral energy Energy of the low pass equivalent signals

12 Cross-correlation Inner Product of two vectors (signals) Hence, we can also write Let s prove that, The complex quantity x,y is called cross-correlation coefficient, given by Two signals are Orthogonal if their inner product (and hence ) is zero. If xl,yl = 0 then x,y = 0. Orthogonality in baseband implies orthogonality in passband but not vice versa.

13 2.2 Signal Space Inner Product Unit Vector - projection of the vector L2 - norm is simply the length of a vector Two vectors are orthonormal iff they are orthogonal and each vector has a unit norm Same vector concepts apply to signals as well. Summation is replaced by integral on to the unit vector ei

14 Orthonormal Bases Let a set of vectors B = {v1, v2 v3..vk} are linearly independent, unit vectors and orthogonal to each other B is called an orthonormal set (orthogonal and normalized) Then any arbitrary vector in that space spanned by a subspace V can be represented as linear combination of all the orthonormal basis B multiplied by different constant terms: X = {c1. v1 + c2. v2 + ck. vk} Think about the R3 euclidean coordinate system we know of Therefore, vk.x = ck (how?) Try it yourself: Given two vectors v1 = {⅓, ⅔, ⅔}T and v2 = {⅔, ⅓, -⅔}T check if these form an orthonormal set Given two vectors v1 = {⅗, ⅘ }T and v2 = {-⅘, ⅗ }T, what is the c matrix for a vector X = {9, -2} Orthonormal bases are good for creating coordinate systems for signals

15 Gram-Schmidt Orthogonalization Method Construct set of orthonormal vectors from a set of n-dimensional vectors, vi The first orthonormal vector is simply, In other words, u1 is the orthonormal vector for subspace with span (v1) Find the unit vector orthogonal to u1 such that In other words, u1, u2 is the orthonormal vector for subspace with span (v1, v2) = span (u1, u2) u2 normalize v2 Similarly, for 3 dimensional span v1 θ u1 u2 u1 normalize v3 u3 Projection of v3 on plane span(v1, v2 ) Since, <v1, v2> = v1 v2 cos θ and v1 = v1 u1 <v, v > projv1 (v2) = v2 cos θ x u1 = 1 2 u1 v1 v1 <u1, v2> or, u1 v1 Projection of v2 on v1

16 GSOM for signals Construct a set of orthonormal waveforms from a given set of finite energy signals {sm(t)} [no. of bases N M (waveforms)] The first basis is same as in vectors, except normalized using the energy The second basis is obtained by projecting s2(t) on to ɸ1(t) Projection The kth orthonormal basis is given by, where, Normalize

17 Example M dimensional Signals N Orthonormal Basis

18 Signal Constellation Once we have constructed the set of orthonormal waveforms {ɸn(t)}, we can express the M signals {sm(t)} as linear combinations of the {ɸn(t)} Therefore, each signal can be represented by a vector, and the collection of M vectors in a N - dimensional signal space is called constellation Vector to Signal Signal to Vector

19 Example

20 2.3 - Random Variables Bernoulli Binomial - Sum of n independent Bernoulli trials with parameter p Uniform Gaussian A Gaussian random variable with m = 0 and σ = 1 is called a standard normal. A function closely related to the Gaussian random variable is the Q function Properties of Q-function Complementary Error Function

21 2.7 Random Processes Let (Ω, F, P) be a probability space. A real random process, N(ω, t), with ω Ω is a single-valued function or mapping from Ω to real valued functions of an index set variable t. The function N(ω1, t) for a fixed ω1 is called a sample path of the random process If N(t) is a Gaussian random process then one sample of this process, N(t s), is completely characterized with the PDF where,

22 Zero Mean Gaussian Process Most common source of noise in communication systems is thermal noise, which has a zero mean If N(t) is a Gaussian random process then two samples of this process, N(t1) and N(t2), are completely characterized with the PDF Sample Variance (zero Mean) Correlation coefficient (Auto) - Correlation function is defined by The variance and correlation coefficient can be expressed in terms of the autocorrelation function

Stationarity If the statistical description of a random process does not change over time, it is said to be a stationary random process Also, the density function describing M samples of the random

23 Stationarity If the statistical description of a random process does not change over time, it is said to be a stationary random process Also, the density function describing M samples of the random process is independent of time shifts in the sample times, i.e., for any value of M and t0 V. IMP - statistical description of any two samples taken from a stationary random process is a function of the time difference between the two time samples This implies E(N(t)) = 0 and σ2n(ti) = σ2n is a constant Also, RN(0, t1) = RN(t0, t1+ t0) for any t0 This implies that the correlation function is essentially a function of only one variable RN( t1, t2) = RN (τ ) = E[ N( t1) N( t1 τ )] τ = t1 - t2

24 Frequency domain Follows from In frequency domain, Since, FT is a complex random function and the energy grows unbounded as measurement interval Tm gets large, we measure power spectral density The average PSD of the random process is given by

25 Frequency Domain PSD of a stationary random process is given by the Wiener-Khinchin theorem Proof: Total Average Power Two useful results

26 AWGN The PSD of AWGN is Where, N0 = KT The Autocorrelation function AWGN is More on simulating AWGN and its properties using MATLAB of

27 Linear systems and Random Process Assume the filter input is W (t ), the filter impulse response is hr(t ), and the filter output is N (t) The random process that results from linear time-invariant filtering of a stationary Gaussian random process is also a stationary Gaussian process The average power of noise at the output of the filter

2.9 BP and LP Random Processes A bandpass process, the PSD is located around frequency (+/- f0) and for lowpass the PSD is located around zero frequency A bandpass process is real, zero mean,

28 2.9 BP and LP Random Processes A bandpass process, the PSD is located around frequency (+/- f0) and for lowpass the PSD is located around zero frequency A bandpass process is real, zero mean, stationary process. Hence I-Q components are Also define a low pass equivalent process (Xi and Xq are both zero mean) PSD of Xi and Xq is given by See Example If the spectrum X(f) is symmetric around f=0 then Sx(f+fo) and Sx(f-fo) are same, i.e, I and Q components of the bandpass random process are uncorrelated (or independent processes)

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