Proc. IEEE Int. Symp. Circuits and Systems (ISCAS) Kobe, Japan 2005 Defining Correlation Functions and Power Spectra for Multirate Random Processes
Charles W. Therrien Department of Electrical and Computer Engineering Naval Postgraduate School Monterey, California 93943–5000 Email: [email protected]
Abstract— This paper proposes a representation for the time- [ W lag cross-correlation function of two random processes sampled at different rates and its related cross-power density spectrum, using the theory of lattices. The representation of autocorrelation and the (auto)power spectral density function then follows directly. W I. INTRODUCTION The time-lag cross-correlation function for two (continuous- \ W time) random processes is defined as ∗ Rxy(t; τ)=E {x(t)y (t − τ)} (1) W while what will be called the “traditional” cross-correlation function is defined as Fig. 1. Continuous random processes sampled at different rates. (Note that ∗ Rxy(t, t )=E {x(t)y (t )} (2) the samples do not line up in time except at points indicated by the dashed lines.) The time-lag form has a number of advantages. Among these are that it is easily related to concepts of the spectrogram, vari- ous stochastic time-frequency distributions (Wigner, Rihaczek) is reasonable to assume that the sampling rates are integer- as well as the time-frequency correlation or “ambiguity” valued. We further assume that the samples “line up” at T T function and cyclic correlation and power density for cyclo- multiples of , where is called the common period and M M stationary (or “periodically correlated”) random processes [1], is defined by the smallest integers x and y satisfying [2]. The time-lag form also leads to a clear distinction between T = MxTx = MyTy (3) random processes that are wide-sense stationary (wss), those that are wide-sense cyclostationary (wscs), and those that are With some abuse of notation, we denote the resulting discrete- x[m ]=x(m T ) y[m ]=x(m T ) periodic. Stationary processes are independent of t, cyclosta- time signals as x x x and y y y m m ∈ Z tionary processes are periodic in t but not in τ, and periodic where x and y (the set of integers). processes are periodic in both variables. The traditional and time-lag forms of the cross-correlation The analysis of random processes sampled at different rates function for the continuous-time random processes are defined from a statistical point of view leads to the concept of joint by (1) and (2) above with the relation cyclostationarity between these processes. The representation τ = t − t (4) for the time-lag cross-correlation function and cross-spectra of t, t,τ ∈ R two such random processes, seems to have eluded researchers The definitions apply for all (the real numbers). in signal processing, however, and has impeded research for It is straightforward and intuitive to define a discrete cross- statistical multirate signal processing. We believe that such correlation function for the two sampled random processes in representation is fundamental to the statistical analysis of the traditional form as ∗ multirate processes and is necessary for further progress in Rxy[mx,my]=E {x[mx]y [my]}=Rxy(mxTx,myTy) the field. This cross-correlation function is defined on a rectangular lat- II. CROSS-CORRELATION tice [3] denoted by ΛR and depicted in Fig. 2. The generating matrix for the lattice is given by Consider two continuous-time random processes x(t) and ⎡ ⎤ y(t) as illustrated in Fig. 1. Beginning at t = t =0, these two ⎣ ⎦ Tx 0 processes are sampled at rates of Fx and Fy samples/second VR = vx vy = 0 Ty with sampling intervals of Tx =1/Fx and Ty =1/Fy.It W where mx,ly ∈ Z. Now, in order to represent the discrete, time-lag correlation function algebraically, observe that the variables t, t and τ are related by t t 10 t = T = 7[ t τ 1 −1 τ (8)
The matrix T is an involutory matrix (T−1 = T). Using (5), Y[ (8), and (7) in that order, produces W mx −1 mx Y\ = V TV m R L l 7\ y y It is easily shown by carrying out the matrix multiplication Fig. 2. Cross-correlation function Rxy[mx,my] defined on rectangular that lattice ΛR. 10 V−1TV = = T R L 1 −1 where vx and vy are the two basis vectors shown in the figure. Thus it is seen from the previous equation that my = mx −ly. A point on the lattice is thus represented by We can therefore define the discrete, time-lag cross-correlation function as t mx = VR = mxvx + myvy (5) t my def ∗ Rxy[mx; l] = E {x[mx]y [mx − l]} (9) Now consider the representation of the time-lag correlation where we have dropped the subscript on the lag variable to function Rxy(t; τ) in discrete time. The variable t is of the simplify notation. form t = mxTx. Also, τ can only take on values corresponding to the actual samples available. To see what values are allowed, The algebraic definition of cross-correlation in the multirate we write (4) as case is seen to be the same as would be used for processes with a single sampling rate, but the interpretation is different. τ = t − t = m T − m T x x y y As seen above, the variables mx and l specify a position on a lattice ΛL. This lattice degenerates to a rectangular lattice Thus for any fixed value of mx, the allowable values of τ only in the case of identical sampling rates (Tx = Ty), where are separated by integer multiples of Ty. The possible values for (t, τ) can be seen to fall on the (non-rectangular) lattice the generating matrix (6) becomes diagonal. Taking a similar approach, one can define the cross- ΛL depicted in Fig. 3. The generating matrix for this lattice correlation function Ryx as t def ∗ Ryx[my; l ] = E {y[my]x [my − l ]}
Observe that this function is defined on its own lattice ΛL and that the lag variable l is distinct from the lag variable l in Rxy. a' b' The relation between these two cross-correlation functions and Tx the traditional forms of cross-correlation is given by
Rxy[mx; l]=Rxy[mx,mx − l] (10) ∗ ∗ =Ryx[mx − l, mx]=Ryx[mx − l; −l] vx a b τ vy All four of the forms are equivalent in the sense that any one Ty of them can be used to specify all the possible correlation values that exist between the two sampled random processes. Fig. 3. Time-lag cross-correlation function Rxy[m; l] represented on a lattice ΛL. III. INDEXED CORRELATION FUNCTION While the representation (9) is quite general, an important is given by ⎡ ⎤ property is not directly apparent. That property has to do with stationarity (or more generally cyclostationarity, if it exists) of ⎣ ⎦ Tx 0 VL = vx vy = the underlying continuous random processes. Refer to Fig. 3 T − T T (6) x y y again, and assume that the random processes x(t) and y(t ) are jointly wide-sense stationary (wss). In this case the continuous- Any point on the lattice can be represented by coordinates time cross-correlation function Rxy(t; τ) is a function of τ t mx alone. Thus Rxy(t; τ) is a surface that resembles a tunnel with = VL = mxvx + lyvy (7) τ ly its axis parallel to the t axis. The contours of equal correlation are straight lines parallel to the t axis. Because the function Although this function is also defined on a lattice, the lattice is constant in t, the periodic sampling of the lattice induces is rectangular, since there is no diversity in the sampling rates. a periodicity in correlation. For example, the points a and The second-moment statistics then are as listed in Table I. The b shown in Fig. 3 have the same value of correlation as the TABLE I points a and b because they are at the same distance from LISTING OF TIME-DOMAIN STATISTICS FOR MULTI-RATE SIGNAL the t axis. Further, observe that this periodicity is not two- PROCESSING. dimensional. The periodicity occurs along only one direction, namely that of the t axis. The periodicity of the cross-correlation function Definition Rate Rxy[mx; ly] is not immediately evident from (9) because the t lattice basis vectors do not align with the axis. It can also µx[m]=E {x[m]}, µy[m]=E {y[m]} Fx, Fy x(t) y(t ) ∗ be shown algebraically, however, as follows. If and Rx[m; l]=E {x[m]x [m − l]} Fx ∗ are jointly wss then (9) can be written as Rxy[m; l]=E {x[m]y [m − l]} mixed ∗ Rxy[mx; ly]=E {x[mx]y [mx − ly]} c = Rxy (mxTx − (mx − ly)Ty) covariance can be defined in an analogous way and is related c to the other quantities as where Rxy is written here with a single argument (τ) which ∗ represents the difference of the sample times. To demonstrate Cx[m; l]=Rx[m; l] − µx[m]µx[m − l] (a) (15) the periodicity, let mx = mx + Mx and ly = ly + Mx − My ∗ Cxy[m; l]=Rxy[m; l] − µx[m]µy[m − l] (b) where Mx and My are defined by (3). Then c Discrete time processes with constant or periodic mean, and Rxy[mx; ly]=Rxy (mxTx − mxTy + lyTy with correlation functions of the form (13) are said to be jointly +MxTx − MxTy +(Mx − My)Ty) wide-sense cyclostationary (wscs).
added terms V. S PECTRAL REPRESENTATION FOR JOINTLY WSS By virtue of (3), the added terms sum to 0 and the periodicity PROCESSES is demonstrated. A. Cross-Power Spectral Density m l For the case of wss processes, the variables x and y can The two-dimensional (2-D) cross-power spectral density therefore be confined to the first period of the lattice (i.e., the function of the continuous signals x(t) and y(t) can be defined m = l =0 period containing the point x y ). That is, given as arbitrary values for the variables mx and ly, we can find ∞ ∞ m l c −j2πft −j2πfτ equivalent values and in the first period of the lattice in Sxy(f ,f)= Rxy(t; τ) e e dt dτ (16) the sense that Rxy[m; l]=Rxy[mx; ly]. To confine the indices −∞ −∞ to the first period, choose m as The following observations can be made: m = m − PM 1) If x(t) and y(t) are jointly wss, then Rxy is constant x x (11) c with respect to t. This implies that Sxy(f ,f) has where P = mx/Mx. In other words, P is the number of support only on the f axis, where it is singular. full periods covered by mx and m ≡ mx mod Mx. Then 2) If x(t) and y(t) are jointly wscs with period P , then c it follows from the arguments in the previous paragraph that Rxy(t; τ) is periodic in t. This implies that Sxy(f ,f) choosing has support on an infinite set of lines f = k · 2π/P for l = ly − P (Mx − My) (12) k ∈ Z. forces Rxy[m; l] and Rxy[mx; ly] to have the same values. These are the two cases of primary interest since wss or wscs The equivalent correlation function defined by processes sampled at different rates become jointly wscs. It follows from the theory of sampling on a lattice (e.g., c Rxy[m; l]=Rxy (m(Tx − Ty)+lTy) (13) [4]) that the corresponding sampled version of Rxy(t; τ) has 0 ≤ m ≤ M − 1 , −∞
ISCUSSION )[ VI. D The representation of the correlation function in the time- X[ lag form (as opposed to the traditional time-time form) has important advantages for the analysis of random processes sampled at different rates. While the latter representation )\ is straightforward, the former has been elusive. The correct X\ definition is clear however, if one considers correlation as represented on a two-dimensional lattice. The corresponding )[)\ I spectral representation fits well with established descriptors for processes exhibiting cyclostationarity. In future publications we will demonstrate how this repre- sentation naturally fits in to the extension of linear optimal D filtering to the multirate case both in time and frequency domains. Some previous publications [5], [6] have addressed Nk these problems without the benefit of the results discussed 0[ here.
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