Theorem 10 (m odd). Let n be an odd prime, m a divisor of n−1, and ω = e2πin . Let A = {a1, ..., am} be the unique subgroup of (Z/nZ)× of sizem, and set U= diag(ωa1, ..., ωam)∈Cm×m, v= √1m[1, ...,1]T ∈Cm×1, andM= [v,Uv, ...,Un−1v]. Setκ:= n−1m .
If mis odd, then the coherence ofM is upper-bounded by
µ≤ 1 κ
s 1
m +κ 2 −1
β 2
+κ 2
2
β2, (3.28)
whereβ=q
1
m κ+m1 .
Proof. We delay the proof of this theorem until Appendix B.1.
It is worth noting that this latter bound has no analog in the κ = 3 situation becausem must always be even in that case. Indeed, ifnis any prime greater than 2 it is necessarily odd, and n−1 is even. Thusm= n−13 is also even. In the Appendix B.1 we will give an alternate classification for exactly when this latter coherence bound applies. This will allow us to apply our bound to a more general class of frames, which we will discuss in the next chapter. We illustrate the upper and lower bounds for κ= 3 in Figure 3.2 and the two upper upper bounds from Theorem 10 for whenκ= 4.
Whenκ= 4, we can also derive different lower bounds on the coherence for whenm is even or odd, and together with the two upper bounds from the theorems they form two non-overlapping regions in which the coherences can fall in the graph. While these regions will exist for everyκ, they will sometimes overlap (that is, the lower bound on coherence for m even could be less than the upper bound formodd).
0 50 100 150 200 250 300 350 400 450 500 0
0.05 0.1 0.15 0.2 0.25
m
Coherence
Upper Bound Lower Bound Welch Bound Actual Coherence
Figure 3.2: The upper and lower bounds on coherence forκ= 3.
0 50 100 150 200 250 300 350 400 450 500
0 0.05 0.1 0.15 0.2 0.25 0.3
m
Coherence
Upper Bound (m even) Lower Bound (m even) Upper Bound (m odd) Lower Bound (m odd) Welch Bound
Actual Coherence (m even) Actual Coherence (m odd)
Figure 3.3: The upper and lower bounds on coherence forκ= 4.
generated by the matrixU, which we define as follows: For any`∈H, letD`= diag
ω`a01, ..., ω`a0m0 , anm0×m0 diagonal matrix with the elements ofω`a0, a0∈A0 along the diagonal. Then defineUto be the (block) diagonal matrix
U:= diag(D`1,D`2, ...,D`d).
Now, since each coset of A0 in H consists only of elements relatively prime to n, then we see that this matrix will indeed maintain the property of having multiplicative ordern, as in our original framework. In fact, if we choose`iA0 =gin−1m A0, for eachi= 1, ..., d, then sincegn−1m is a generator forA, we find that the cosets{`1A0, ..., `dA0}are precisely the cosets ofA0 as a subgroup ofA. These cosets partition the elements ofA, so we retrieve the matrix obtained from our original construction, withU= diag (ωa1, ..., ωam) up to a permutation of the elements ofA(which will not affect the values of the inner products in (3.15)). Thus, this new construction is a direct generalization of our original work. Another special case is whenA0= 1, the trivial subgroup. In this case, selecting cosets for A0 is nothing more than selecting individual rows of then×nFourier matrix forM, with the exception of the row of all 1’s.
As we cycle through the powers of U, eachD`i cycles through the different cosets ofA0 in some order. Since some powers ofUmay give rise to permutations of the same cosets, and hence lead to the same corresponding inner product from Equation 3.24, it can take some care to determine precisely how many distinct inner products we have in our constructed matrix. We know that it can be as few as n−1m , as is the case when the chosen cosets of A0 partition A. In general, we have the following theorem:
Theorem 11. Letnbe a prime,ma divisor ofn−1, andm0 a divisor ofm, withm=dm0. LetAbe the unique subgroup ofH= (Z/nZ)× of sizem, andA0={a01, ..., a0m0}the unique subgroup of sizem0. Let{`1A0, ..., `dA0}be a set ofdcosets ofA0 inH. Setω=e2πi/n andv= √1m[1, ...,1]T ∈Cm×1, and form the matrices D`i = diag
ω`ia01, ..., ω`ia0m0
∈ Cm
0×m0, U := diag(D`1,D`2, ...,D`d) ∈ Cm×m, and M = [v,Uv, ...,Un−1v] ∈ Cm×n. Then M has at most d·(n−1)m distinct values of the inner products between its columns.
Proof. We know that the distinct inner products between the normalized columns ofMwill correspond to the powers ofU. Thebthpower ofUcan be written asUb = diag Db`
1, ...,Db`
d
. Thus, the inner product corresponding to this power ofUis
1 m
X
a0∈A0
ω`1(ba0)+ X
a0∈A0
ω`2(ba0)+...+ X
a0∈A0
ω`d(ba0)
. (3.29)
Thus, there can only be as many such sums as there are cosetsbA0. Since there are n−1m0 = d·(n−1)m
cosets ofA0 in H, we have our result.
This coset construction offers us a tradeoff. By using the smaller groupA0 (of size md) to construct our matrix as opposed toA (of sizem), we gain the possibility of having nice cancelation properties among the sums P
a0∈A0ω`i(ba0) in (3.29) at the cost of having more inner products to control. But since the number of distinct inner products can increase only by a factor ofdat most, this can turn out to be a worthwhile tradeoff, and indeed we have examples where we can strictly decrease the coherence ofMby using this construction. (See Fig. 3.4).
We can now can formulate the problem of constructing low-coherence matrices as an optimization problem, where we can optimize over both the choice ofA0 and the set of cosets{`1A0, ..., `dA0}. For fixedmandn, wherenis a prime andma divisor ofn−1, we must solve the following:
min
m0|m, `∈G×mm0
maxb∈H
1 m
m/m0
X
i=1
X
a∈Am0
ω`i(ba)
, (3.30)
whereH = (Z/nZ)×,`= (`1, ..., `d),H×mm0 denotes the Cartesian product ofH with itself mm0 times, andAm0 denotes the unique subgroup ofH of sizem0.
In practice, it is typically not feasible to perform this exact optimization since it requires a search over the latticeH×mm0 for everym0dividingm. One simple way to deal with this problem is to fixm0 and randomly sample`∈H×mm0 to search for the smallest value of the objective function. Note that ifm02|m01 (or equivalently, Am02 ≤ Am01), then searching over cosets of Am02 encompasses the search over cosets of Am01 since we can express Am01 as a union of cosets ofAm02. One might therefore be tempted to argue that it is unnecessary to search over cosets ofAm01 at all. There is, however, value in searching over these cosets, since this search will converge to its optimal value much faster than the search over cosets of the smaller group. See Figure 3.4.
Of course, we can still bound the coherence of the frames that can arise from this construction using that of our previous frames.
Theorem 12. Let n be a prime, m a divisor of n−1, m0 a divisor of m, and d = mm0. Let A0 = {a01, ..., a0m0} be the unique subgroup of H := (Z/nZ)× of order m0. Set ω =e2πi/n and v =
√1
m[1, ...,1]T ∈Cm×1, and as before choose a set of cosets{`1A0, ..., `dA0}ofA0inH. Form the matri- cesD` = diag
ω`a01, ..., ω`a0m0
∈Cm
0×m0 for any `∈H, andU:= diag(D`1,D`2, ...,D`d)∈Cm×m. If M1 = [v,Uv, ...,Un−1v] ∈Cm×n has coherence µ1 and M2 = [v,D1v, ...,Dn−11 v] ∈ Cm
0×n has coherenceµ2, then we haveµ1≤µ2.
Proof. The result comes from a simple application of the triangle inequality: from Equation (3.29),
0 200 400 600 800 1000 0.16
0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32
Iteration
Lowest Coherence Observed
m’ = 1 m’ = 14 m’ = 70
Figure 3.4: Randomly sampling ` to search for the optimal coherence over cosets of subgroups of sizem0 for various values ofm0 (= 1, 14, 70). (Plot shows the lowest coherence found up to a given iteration). Here,n = 491, and m= 70. The figure shows that m0 = 14 quickly achieves the lowest values of coherence.
we see that
µ1= max
b
1 m
X
a0∈A0
ω`1(ba0)+...+ X
a0∈A0
ω`d(ba0)
(3.31)
≤ 1
dm0 max
b1
X
a0∈A0
ω(`1b1)a0
+...+ max
bd
X
a0∈A0
ω(`dbd)a0
!
(3.32)
= 1
d d·max
s
1 m0
X
a0∈A0
ωsa0
!
(3.33)
=µ2. (3.34)
Theorem 12 naturally allows us to use the bounds from Theorems 9 and 10 to explicitly bound the coherence from our coset optimization in terms ofr, m and d, though it is worth noting that in practice we achieve coherence significantly lower than these bounds.