We take the case for Nb = 3 as an example without loss of generality. Assume m = 1, then: the vector bi and the elements in F are all of unit magnitude, the elements of gi must have identical magnitude. In order to make the explanation clear, we denote
G = [g0 g1g2] =
The relation between bi’s and gi’s is showed in Fig. 3.1. We proceed by invoking a
˃
technique similar to (2.15). Performing the rate expanding on the time-domain vectors
g0, g1, and g2 by inserting some zeros with an expanding rate of Nb = 3, we obtain the
After taking Nb2-DFT on the expanded vectors(sequences), the corresponding frequency-domain sequences will repeat periodically. Then we do some time-shifting on ge1 and
ge2 such that there is no overlapping if we sum ge0, ge1, and ge2 together. Denote the time-shifted vectors as ges0, ges1, and ges2, and then
ges0 = [ g0 0 0 g1 0 0 g2 0 0 ]T (3.8)
ges1 = [ 0 g3 0 0 g4 0 0 g5 0 ]T (3.9)
ges2 = [ 0 0 g6 0 0 g7 0 0 g8 ]T. (3.10) This shifting operation will cause some phase rotation in frequency-domain. In fact the phase rotation is not important, and we will show the reason later. We depict these steps in Fig. 3.2 in which the phase rotation effects are also shown.
Summing the vectors ges0, ges1, and ges2 together, we obtain
g = ges0+ ges1+ ges2 = vec(GT). (3.11) We plot the corresponding result in Fig. 3.3. The corresponding Nb2-DFT points of g are all with the same magnitude.
In order to explain the effects of the phase rotation, we need to recall some properties introduced in chapter 2. Denote the AC function of a sequence x(n) as θ(x, x)(n) and its Nc-DFT as Θ(x, x)(k), we hope that
θ(x, x)(n) = Ncδ(n), (3.12)
and
Θ(x, x)(k) = X(k)X∗(k) = Nc. (3.13) These two equations had been mentioned in (2.16) and (2.17). Hence if we set X(k) =
√Ncejφk, i.e., all frequency components have the same magnitude, then (3.13) can be
achieved. The constant √
Nc can be ignored since it does not affect the AC and CC properties. Notice that there is no constraint on the choice of {φk}. We can conclude
that a sequence has perfect periodic AC property if all of its frequency components have the same magnitude. Hence the new sequence g has perfect AC property. This statement also explains why the phase rotation induced from the time-shifting operation is not important. From Figs. 3.2 and 3.3, we also can get the idea why the transpose and the stacking operations in (3.2) are needed. Up to now, we have concentrate our
˃
Figure 3.2: Rate-expanding and time-shifting. The time-domain sequences and their corresponding frequency-domain sequences are showed.
̇
Figure 3.3: Generate the new sequence with perfect periodic AC property. All frequency components are of equal magnitude.
discussion on the case m = 1 only. The case m= 1 is the subject of our next discourse.
We need to re-define some notations.
Let Xi(k) = biδ(k− i), then [Xi(0), Xi(1), . . . , Xi(Nb− 1)]T = bi. Taking Nb-point IDFT on X(k)’s, we get a sequence xi(n), for 0≤ n ≤ Nb−1, and [xi(0), xi(1), . . . , xi(Nb− 1)]T = gi(m=1) gi(1). This is equivalent to the case discussed before for m = 1. The ith column vector of the matrix G(m) with m= 1 can be written as:
gi(m)= [x(m)i (0), x(m)i (1), . . . , x(m)i (Nb− 1)]T, (3.14) phase-rotated version of xi(n). Hence we have following relations:
xi(n)DF T−→ Xi(k) = biδ(k− i) =⇒ x(m)i (n) DF T−→ Xi[(k− i(m − 1))Nb], (3.16) and Xi[(k− i(m − 1))Nb] = biδ[(k− im)Nb]. For convenience, we will use the notation (n)Nb to denote (n modulo Nb). The term biδ[(k − im)Nb] means that the effect of
different values of m is simply the step size of index shifting in frequency-domain. If we can keep the shifted tones still non-overlapping, the sequence vec{G(m)T} can preserve perfect periodic AC property. To change another words, if there are some values of m such that the sequence vec{G(m)T} has perfect AC function, then we can find a column-reordered matrix P from the diagonal matrix D({bi}) = [b0, . . . ,bNb−1], such that F(Nb,−m)D({bi}) = F(Nb,−1)P. The RHS is the case that m = 1, which has been discussed. Now we check what the values of m should be such that the shifted tones are still non-overlapping. We consider two cases separately, depending on if m is prime to Nb or not.
Case I g.c.d.(m, Nb) = 1
If there is one tone lapped over another after being shifted, then
im mod Nb= jm mod Nb, for i= j, 0 ≤ i, j < Nb.
⇒ (i − j)m mod Nb = 0
⇒ (i − j)m = pNb, p∈ Z (3.17)
The assumption that g.c.d.(m, Nb) = 1 implies that Nb|(i − j), ∴ i = j. We thus conclude that if there are two tones with i= j, they will not overlap after shifting.
Example 1 Consider the case, Nb = 4, m = 3, W4 = ej2π/4, and b ={W41, W42, W43, W44}.
In this case we have g.c.d.(m, Nb) = 1. The basic orthogonal sequence matrix G is de-fined by F(Nb,−m)D({bi}). Taking DFT on each column vector of G(m), we get the column vectors of P . The result is shown in Fig. 3.4, and the AC function of the sequence vec(G(m)T) is plotted in Fig. 3.5.
Case II g.c.d.(m, Nb) = d:
Let m = hd, Nb = kd, and g.c.d.(h, k) = 1.
im mod Nb = jm mod Nb
⇒ (i − j)hd mod kd = 0
⇒ (i − j)h mod k = 0
⇒ (i − j)h = qk, q ∈ Z (3.18)
g.c.d.(h, k) = 1 ⇒ k|(i − j) ∴ i = j mod k.
Hence there will be d tones lapped together. We can denote {j, j + k, j + 2k, . . .} as a coset. There will be d elements in this coset and one can find N/d different cosets.
From the above discussion we conclude that the sequence vec{G(m)T} has perfect periodic AC property if g.c.d.(m, Nb) = 1.
Example 2 Consider the case–Nb = 4, m = 2, W4 = ej2π/4, and b ={W41, W42, W43, W44}.
In this case we have g.c.d.(m, Nb) = 2. The basic orthogonal sequence matrix G(m) is defined by F(Nb,−m)D({bi}). Taking DFT on each column vector of vec(G(m)T), we obtain the desired result as shown in Fig. 3.6. We notice that if we sum up all column vectors, there must be some tones overlapping each other. Hence the AC function will not be perfect; see Fig. 3.7.
The above discussion indicates that the sequence vec(G(m)T) has perfect AC function if g.c.d.(m, Nb) = 1. Furthermore, if the bi’s are of the same magnitude, the generated sequence will be composed of complex numbers with the same magnitude. Observing Fig. 3.3, we can notice that the elements of the sequence have the same magnitude both in time-domain and frequency-domain. As mentioned before, a sequence has perfect AC function if all of its frequency components have the same magnitude. Hence if we exchange the roles of the “time-domain” sequence and the “frequency-domain” sequence, the AC property still can be maintained. Both the sequence generated in this chapter and the FZC sequence have this property. With this property, the step of (2.10) can be
modified as
Ci(λ) =
Kxk ; λ = Kk + i,
0 ; otherwise. (3.19)
Then the reduced sequence generation procedure is identical to that of the PS sequence.
0 0.5 1 1.5 2 2.5 3
Figure 3.4: g.c.d.(m, Nb) = 1. Magnitude plot for the DFT of each column vector of G(m).
AutoCorrelation function for the basis sequence with m=3
Figure 3.5: g.c.d.(m, Nb) = 1. AC function of the sequence of m = 3.
0 0.5 1 1.5 2 2.5 3
Figure 3.6: g.c.d.(m, Nb) = 1. Magnitude plot for the DFT of each column vector of G(m).
AutoCorrelation function for the basis sequence with m=2
Figure 3.7: g.c.d.(m, Nb)= 1. AC function of the sequence of m = 2.
Chapter 4
Multi-dimensional Arrays
Like the one dimensional (1D) case, two dimensional (2D) arrays that possess some desired AC or CC properties are useful in sonar/radar and multimedia applications.
Similarly, higher dimensional array signal are needed in some cognitive radio and com-puter graphics. In this chapter, we extend the concepts developed for one-dimensional sequences to two or higher dimensions cases. The notations and definitions used here follow those of [9].
4.1 Array correlation functions
Let an array sequence A = ai,j be denoted by
A =
The two-dimensional periodic AC function between two array sequences A and B having the same dimensions is defined as
RA,B(φ, ω) =
An array is called perfect array if its periodic AC function satisfies
RA,A(φ, ω) = RA(φ, ω) =
E, (φ, ω) = (0, 0)
0, (φ, ω)= (0, 0) , (4.3)
where E =N1−1
p=0 N2−1
q=0 |ap,q|2.
There are many earlier works on the syntheses of perfect arrays. We will apply one of the synthesis methods introduced in [9] to obtain a perfect array. This method is based on
To begin with, we need a perfect array sequence. This sequence will be referred to as the basis array. We apply the folding method to the FZC sequence of length N1N2, where gcd(N1, N2) = 1, and then we get an N1× N2 perfect array. Taking the two-dimensional DFT on this basis array, we obtain
F (u, v) = corresponding two-dimensional DFT’s are F(i)(U, V ) defined by
F(i)(U, V ) =
This assignment is illustrated in Fig. 4.1. Taking the two-dimensional IDFT on F(i)(U, V ), we obtain an array sequence C(i) of dimension K1N1 × K2N2, where the