Real-time time encoding and decoding machines
Methods for decoding a signal encoded by a Time Encoding Machine (TEM) include defining a plurality of time-windows, each time-window corresponding to a portion of a TEM-encoded signal and made up of a plurality of trigger values, at least two of the time-windows overlapping, decoding each of the time-windows using a Time Decoding Machine (TDM) to generate a decoded time-window, and stitching the decoded time-windows together to generate a TEM-decoded signal.
1. A circuit for decoding a signal encoded by a Time Encoding Machine (TEM) comprising:
a Time Decoding Machine (TDM) for decoding a portion of a TEM-encoded signal, the TEM-encoded signal comprising a plurality of action potentials encoded in a time sequence, said TDM comprising a TDM-input and a TDM-output, said TDM-input for receiving the portion of the TEM-encoded signal over a first time-window, said TDM-output for transmitting a decoded approximated signal over said first time-window; and
a stitching circuit, having a stitching-input connected to said TDM-output, said stitching circuit combining said decoded approximated signal over said first time-window with at least one other decoded approximated signal over a second time-window, wherein said first time-window and said second time-window overlap.
2. The circuit of claim 1 wherein said TDM calculates a value of said TEM-encoded signal, u, over said first time-window, [t i , t i+N ], wherein the value of u at a given time, t, is approximated by solving for periodic bandlimited signal
u
i
(
t
)
=
∑
n
=
0
N
j
(
Ω
-
n
2
Ω
N
)
d
i
,
n
ⅇ
j
(
-
Ω
+
n
2
Ω
N
)
t
;
where bandwidths of each period of u i (t) are Ω and 2Nπ/Ω (for N≧1), respectively;
coefficients [d i ] n =d i,n , and d i is solved by letting z n m =e jm2Ωi i+n /N where n, m=0,1, . . . , N, and solving for a vector b such that
for n=0, . . . , N−2 do:
for m=N, . . . , n+1 do:
b m =( b m −b m−1 )/( z m −z −m−1 )
for n=N−2, . . . , 0 do:
for m=n, . . . , N−2 do:
b m =n m −b m+1 z n
and d i =b.
3. The circuit of claim 2 wherein
d
i
=
b
(
x
i
-
1
α
i
y
i
y
i
H
x
i
)
,
for α i =y i H y i and x i and y i denote solutions of Vandermonde systems V i x i =D i (P−ab H )r i and V i y i =D i a i respectively;
where [V i ] nm =e jm2Ωi i+n /N is a Vandermonde matrix,
D i =diag(e jΩi i+n ) is a diagonal matrix,
P is an upper triangular matrix with values [P] nm =1 and [P] nm =0 for n<m+1 and n≧m+1,
[r i ] n =(−1) i+n+1 T i+n
a H =[ . . . , 0,1, 0, 1], and
b H =[0, . . . , 0, 0, 1].
4. The circuit of claim 2 wherein said coefficients [d i ] n =d i,n are recovered by solving equation V i H V i d i =V i H D i Pq i , where V i H is a conjugate transposition of V i ,
[V i ] nm =e jm2Ωi i+n /N is a Vandermonde matrix,
D i =diag(e jΩi i+n ) is a diagonal matrix,
P is an upper triangular matrix with values [P] nm =1 and [P] nm =0 for n<m+1 and n≧m+1, respectively and
[q i ] n =q l +n, for all n, m=0, . . . , N.
5. The circuit of claim 2 , wherein said first and second time-windows, ω n , are defined by a function
ω
n
(
t
)
=
{
0
if
t
∉
(
τ
n
,
σ
n
+
1
]
,
θ
n
(
t
)
if
t
∈
(
τ
n
,
σ
n
]
,
1
if
t
∈
(
σ
n
,
τ
n
+
1
]
,
1
-
θ
n
+
1
(
t
)
if
t
∈
(
τ
n
+
1
,
σ
n
+
1
]
,
where K represents a value of desired overlapping trigger times, J=N−2M−K, τ n = nJ+M , σ=t nJ+M+K ,N represents a number of trigger times and
θ
n
(
t
)
=
sin
2
(
π
2
·
t
-
τ
n
σ
n
-
τ
n
)
.
6. The circuit of claim 5 , wherein said stitching circuit combines said first and second time-windows by solving equation
u
^
(
t
)
=
∑
n
∈
Z
ω
n
(
t
)
u
nJ
(
t
)
.
7. The circuit of claim 1 further comprising a post-filtering circuit connected to an output of said stitching circuit for receiving a combined TEM-decoded signal, said post-filtering circuit removing an approximate error, e[k]=u(kS)−û(kS), from said combined TEM-decoded signal of said stitching circuit by solving for an equation
e
ap
[
k
]
=
ω
n
(
kS
)
e
nJ
ap
(
kS
)
,
where S represents a sampling rate of said TEM-encoded signal.
8. A method for decoding a signal encoded by a Time Encoding Machine (TEM) comprising:
defining a plurality of time-windows, each time-window corresponding to a portion of a TEM-encoded signal and comprising a plurality of trigger values, the TEM-encoded signal comprising a plurality of action potentials encoded in a time sequence, at least two of said time-windows overlapping;
decoding each of said time-windows using a Time Decoding Machine (TDM) to generate a decoded time-window; and
stitching said decoded time-windows together to generate a TEM-decoded signal.
9. The method of claim 8 wherein each of said time-windows overlaps with at least one other time-window.
10. The method of claim 8 wherein said TDM calculates a value of said TEM-encoded signal, u, over one of said plurality of time-windows, [t i , t i+N ], where the value of u at a given time, t, is approximated by solving for periodic bandlimited signal
u
i
(
t
)
=
∑
n
=
0
N
j
(
Ω
-
n
2
Ω
N
)
d
i
,
n
ⅇ
j
(
-
Ω
+
n
2
Ω
N
)
t
;
where bandwidths of each period of u l (t) are Ω and 2Nπ/Ω (for N≧1), respectively;
coefficients [d i ] n and d i can be solved by letting z n m =e jm2Ωi i+n /N where n, m=0,1, . . . N, and solving for a vector b such that
for n=0, . . . ,N−2 do:
for m=N do:
b m =( b m −b m−1 )/( z m −z m−n−1 )
for n=N−2, . . . , 0 do:
for m=n, . . . , N−2 do:
b m =b m −b m+ 1 z n
and d i =b.
11. The method of claim 10 wherein
d
i
=
b
(
x
i
-
1
α
i
y
i
y
i
H
x
i
)
,
for α i =y i H y i and x i and y i denote solutions of Vandermonde systems V i x i =D i (P−ab H )r i and V i y i =D i a i respectively;
where [V i ] nm is a Vandermonde matrix,
D i =diag(e jΩi i+n ) is a diagonal matrix,
P is an upper triangular matrix with values [P] nm =1 and [P] nm =0 for n<m+1 and n≧m+1,
[r i ] n =(−1) i+n+1 T i+n ,
a H =[ . . . , 0,1, 0, 1], and
b H =[0, . . . , 0, 0,1].
12. The method of claim 10 wherein said coefficients [d i ] n =d i,n are recovered by solving equation V i H V i d i =V i H D i Pq i , where V i H is a conjugate transposition of V i ,
[V i ] nm =e jm,2Ωi i+n /N is a Vandermonde matrix,
D i =diag(e jΩi i+n ) is a diagonal matrix,
P is an upper triangular matrix with values [P] nm =1 and [P] nm =0 for n<m+1 and n≧m+1, respectively and
[q i ] n =q i +n, for all n, m=0, N.
13. The method of claim 10 , wherein said time-windows, ω n , are defined by a function
ω
n
(
t
)
=
{
0
if
t
∉
(
τ
n
,
σ
n
+
1
]
,
θ
n
(
t
)
if
t
∈
(
τ
n
,
σ
n
]
,
1
if
t
∈
(
σ
n
,
τ
n
+
1
]
,
1
-
θ
n
+
1
(
t
)
if
t
∈
(
τ
n
+
1
,
σ
n
+
1
]
,
where K represents a value of desired overlapping trigger times, J=N−2M−K, τ n =t nJ+M , σ″=t nJ+m+K ,N represents a number of trigger times and
θ
n
(
t
)
=
sin
2
(
π
2
·
t
-
τ
n
σ
n
-
τ
n
)
.
14. The method of claim 13 , wherein said stitching comprises adding said time-windows by solving equation
u
^
(
t
)
=
∑
n
∈
Z
ω
n
(
t
)
u
nJ
(
t
)
.
15. The method of claim 8 further comprising post-filtering said TEM-decoded signal by removing an approximate error, e[k]=u(kS)−û(kS), from said TEM-decoded signal by solving for an equation
e
ap
[
k
]
=
ω
n
(
kS
)
e
nJ
ap
(
kS
)
,
where S represents a sampling rate of said TEM-encoded signal.