Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Second Proof of W-K-S Sampling Poisson Summation Formula (PSF) � � � σ δ n σ = δ n /σ . n ∈ Z n ∈ Z If f ∈ PW Ω and σ ≤ 1 / 2Ω, Second Proof : �� � f ( ω − n � � · χ [ − 1 /σ, 1 /σ ) ( ω ) . f ( ω ) = σ ) n ∈ Z �� � �� � � � f ( ω − n � � � · χ [ − 1 /σ, 1 /σ ) ( ω ) = · χ [ − 1 /σ, 1 /σ ) ( ω ) f ( ω ) = σ ) δ n /σ f n ∈ Z n ∈ Z ��� � � ( PSF ) ⇐ ⇒ f ( t ) = σ δ n σ f ∗ sinc σ ( t ) . n ∈ Z Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Errors in W-K-S Sampling Truncation Error : N � f ( n σ )sin( 2 π σ ( t − n σ )) f N ( t ) = σ . π ( t − n σ ) n = − N Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Errors in W-K-S Sampling Truncation Error : N � f ( n σ )sin( 2 π σ ( t − n σ )) f N ( t ) = σ . π ( t − n σ ) n = − N L 2 error � E N = � f − f N � 2 | f ( n σ ) | 2 . 2 = σ | n | > N Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Errors in W-K-S Sampling Truncation Error : N � f ( n σ )sin( 2 π σ ( t − n σ )) f N ( t ) = σ . π ( t − n σ ) n = − N L 2 error � E N = � f − f N � 2 | f ( n σ ) | 2 . 2 = σ | n | > N Pointwise error E N = sup | f ( t ) − f N ( t ) | ≤ ( σ E N ) 1 / 2 . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Errors in W-K-S Sampling, Cont’d Aliasing Error - Let Ω = 1, σ ≫ 1 / 2. � � � 1 / 2 � � � ◦ ( ω ) e 2 π it ω d ω � ( � � | � E A = sup � f ( t ) − f ) � ≤ 2 f ( u ) | du . − 1 / 2 | u |≥ 1 / 2 Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Errors in W-K-S Sampling, Cont’d Aliasing Error - Let Ω = 1, σ ≫ 1 / 2. � � � 1 / 2 � � � ◦ ( ω ) e 2 π it ω d ω � ( � � | � E A = sup � f ( t ) − f ) � ≤ 2 f ( u ) | du . − 1 / 2 | u |≥ 1 / 2 Jitter Error : If sample values are not measured at intended points, we can get jitter error E J . Let { ǫ n } denote the error in the n th sample point. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Errors in W-K-S Sampling, Cont’d Aliasing Error - Let Ω = 1, σ ≫ 1 / 2. � � � 1 / 2 � � � ◦ ( ω ) e 2 π it ω d ω � ( � � | � E A = sup � f ( t ) − f ) � ≤ 2 f ( u ) | du . − 1 / 2 | u |≥ 1 / 2 Jitter Error : If sample values are not measured at intended points, we can get jitter error E J . Let { ǫ n } denote the error in the n th sample point. First we note that if f ∈ PW (1), then, by Kadec’s 1/4 Theorem , the set { n ± ǫ n } n ∈ Z is a stable sampling set if | ǫ n | < 1 / 4. Moreover, this bound is sharp. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Errors in W-K-S Sampling, Cont’d Aliasing Error - Let Ω = 1, σ ≫ 1 / 2. � � � 1 / 2 � � � ◦ ( ω ) e 2 π it ω d ω � ( � � | � E A = sup � f ( t ) − f ) � ≤ 2 f ( u ) | du . − 1 / 2 | u |≥ 1 / 2 Jitter Error : If sample values are not measured at intended points, we can get jitter error E J . Let { ǫ n } denote the error in the n th sample point. First we note that if f ∈ PW (1), then, by Kadec’s 1/4 Theorem , the set { n ± ǫ n } n ∈ Z is a stable sampling set if | ǫ n | < 1 / 4. Moreover, this bound is sharp. � � ��� ∞ � � � � � � E J = sup � f ( t ) − σ n = −∞ δ n σ ± ǫ n f ∗ sinc σ ( t ) � . If we assume | ǫ n | ≤ J ≤ min { 1 / (4Ω) , e − 1 / 2 } , E J ≤ KJ log(1 / J ) , where K is a constant expressed in terms of � f � ∞ and � f ′ � ∞ . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method Adaptive frequency band and ultra-wide-band systems require either rapidly changing or very high sampling rates. These rates stress signal reconstruction in a variety of ways. Clearly, sub-Nyquist sampling creates aliasing error, but error would also show up in truncation, jitter and amplitude, as computation is stressed. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method Adaptive frequency band and ultra-wide-band systems require either rapidly changing or very high sampling rates. These rates stress signal reconstruction in a variety of ways. Clearly, sub-Nyquist sampling creates aliasing error, but error would also show up in truncation, jitter and amplitude, as computation is stressed. Truncation loses the energy in the lost samples. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method Adaptive frequency band and ultra-wide-band systems require either rapidly changing or very high sampling rates. These rates stress signal reconstruction in a variety of ways. Clearly, sub-Nyquist sampling creates aliasing error, but error would also show up in truncation, jitter and amplitude, as computation is stressed. Truncation loses the energy in the lost samples. Aliasing introduces ambiguous information in the signal. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method Adaptive frequency band and ultra-wide-band systems require either rapidly changing or very high sampling rates. These rates stress signal reconstruction in a variety of ways. Clearly, sub-Nyquist sampling creates aliasing error, but error would also show up in truncation, jitter and amplitude, as computation is stressed. Truncation loses the energy in the lost samples. Aliasing introduces ambiguous information in the signal. Increased likelihood of jitter error and unstable sampling sets. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method Adaptive frequency band and ultra-wide-band systems require either rapidly changing or very high sampling rates. These rates stress signal reconstruction in a variety of ways. Clearly, sub-Nyquist sampling creates aliasing error, but error would also show up in truncation, jitter and amplitude, as computation is stressed. Truncation loses the energy in the lost samples. Aliasing introduces ambiguous information in the signal. Increased likelihood of jitter error and unstable sampling sets. Computation is stressed. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d We have developed a sampling theory for adaptive frequency band and ultra-wide-band systems – The Projection Method . Two of the key items needed for this approach are : Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d We have developed a sampling theory for adaptive frequency band and ultra-wide-band systems – The Projection Method . Two of the key items needed for this approach are : Quick and accurate computations of Fourier coefficients, which are computed in parallel. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d We have developed a sampling theory for adaptive frequency band and ultra-wide-band systems – The Projection Method . Two of the key items needed for this approach are : Quick and accurate computations of Fourier coefficients, which are computed in parallel. Effective adaptive windowing systems. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d We have developed a sampling theory for adaptive frequency band and ultra-wide-band systems – The Projection Method . Two of the key items needed for this approach are : Quick and accurate computations of Fourier coefficients, which are computed in parallel. Effective adaptive windowing systems. The Projection Method is also extremely efficient relative the Power Game discussed by Vetterli et. al. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Let f ∈ PW (Ω). For a block of time T , let � f ( t ) χ [( k ) T , ( k +1) T ] ( t ) . f ( t ) = k ∈ Z Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Let f ∈ PW (Ω). For a block of time T , let � f ( t ) χ [( k ) T , ( k +1) T ] ( t ) . f ( t ) = k ∈ Z If we take a given block f k ( t ) = f ( t ) χ [( k ) T , ( k +1) T ] ( t ), we can T − periodically continue the function, getting ( f k ) ◦ ( t ) = ( f ( t ) χ [( k ) T , ( k +1) T ] ( t )) ◦ . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Let f ∈ PW (Ω). For a block of time T , let � f ( t ) χ [( k ) T , ( k +1) T ] ( t ) . f ( t ) = k ∈ Z If we take a given block f k ( t ) = f ( t ) χ [( k ) T , ( k +1) T ] ( t ), we can T − periodically continue the function, getting ( f k ) ◦ ( t ) = ( f ( t ) χ [( k ) T , ( k +1) T ] ( t )) ◦ . Expanding ( f k ) ◦ ( t ) in a Fourier series, we get � � ( f k ) ◦ ( t ) = ( f k ) ◦ [ n ] exp (2 π int / T ) . n ∈ Z Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d � � ( f k ) ◦ ( t ) = ( f k ) ◦ [ n ] exp (2 π int / T ) n ∈ Z � ( k +1) T ( f k ) ◦ [ n ] = 1 � f ( t ) exp ( − 2 π int / T ) dt . T ( k ) T Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d � � ( f k ) ◦ ( t ) = ( f k ) ◦ [ n ] exp (2 π int / T ) n ∈ Z � ( k +1) T ( f k ) ◦ [ n ] = 1 � f ( t ) exp ( − 2 π int / T ) dt . T ( k ) T The original function f is Ω band-limited. However, the truncated block functions f k are not. Using the original Ω band-limit gives us a lower bound on the number of non-zero Fourier coefficients � ( f k ) ◦ [ n ] as follows. We have n T ≤ Ω , i . e . , n ≤ T · Ω . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Choose N = ⌈ T · Ω ⌉ , where ⌈·⌉ denotes the ceiling function. For this choice of N , we compute Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Choose N = ⌈ T · Ω ⌉ , where ⌈·⌉ denotes the ceiling function. For this choice of N , we compute � f ( t ) χ [( k ) T , ( k +1) T ] ( t ) f ( t ) = k ∈ Z � � � ( f k ) ◦ ( t ) χ [( k ) T , ( k +1) T ] ( t ) = k ∈ Z � n = N � � � � χ [( k ) T , ( k +1) T ] ( t ) . ( f k ) ◦ [ n ] exp (2 π int / T ) ≈ f P = k ∈ Z n = − N Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d This process allows the system to individually evaluate each piece and base its calculation on the needed bandwidth. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d This process allows the system to individually evaluate each piece and base its calculation on the needed bandwidth. Instead of fixing T , the method allows us to fix any of the three while allowing the other two to fluctuate. From the design point of view, the easiest and most practical parameter to fix is N . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d This process allows the system to individually evaluate each piece and base its calculation on the needed bandwidth. Instead of fixing T , the method allows us to fix any of the three while allowing the other two to fluctuate. From the design point of view, the easiest and most practical parameter to fix is N . For situations in which the bandwidth does not need flexibility, it is possible to fix Ω and T by the equation N = ⌈ T · Ω ⌉ . However, if greater bandwidth Ω is need, choose shorter time blocks T . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Suppose that the signal f ( t ) has a band-limit Ω( t ) which changes with time. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Suppose that the signal f ( t ) has a band-limit Ω( t ) which changes with time. Change effects the time blocking τ ( t ) and the number of basis elements N ( t ). Let Ω( t ) = max { Ω( t ) : t ∈ τ ( t ) } . At minimum, � ( f k ) ◦ [ n ] is non-zero if n τ ( t ) ≤ Ω( t ) or equivalently, n ≤ τ ( t ) · Ω( t ) . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Let N ( t ) = ⌈ τ ( t ) · Ω( t ) ⌉ . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Let N ( t ) = ⌈ τ ( t ) · Ω( t ) ⌉ . Let f , � f ∈ L 2 ( R ) and f have a variable but bounded band-limit Ω( t ). Let τ ( t ) be an adaptive block of time. Given τ ( t ), let Ω( t ) = max { Ω( t ) : t ∈ τ ( t ) } . Then, for N ( t ) = ⌈ τ ( t ) · Ω( t ) ⌉ , f ( t ) ≈ f P ( t ) , where � � N ( t ) � � � ( f k ) ◦ [ n ] e (2 π int /τ ) χ [ k τ, ( k +1) τ ] ( t ) . f P ( t ) = k ∈ Z n = − N ( t ) Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method Adaptive Windowing Systems Projection Revisited Binary Signals and Walsh Functions Signal Adaptive Frame Theory Projection Method, Cont’d Let f ∈ PW (Ω) and let T be a fixed block of time. Then, Problem : for N = ⌈ T · Ω ⌉ , � � ∞ � N ( f k ) ◦ [ n ] exp (2 π i ( k − 1 2) T )( ω − n � � f P ( ω ) = T ) k = −∞ n = − N � �� sin( π ( ω T 2 + n 2 )) . π ( ω + n T ) Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems General method for segmenting Time-Frequency ( R − � R ) space. The idea is to cut up time into segments of possibly varying length, where the length is determined by signal bandwidth. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems General method for segmenting Time-Frequency ( R − � R ) space. The idea is to cut up time into segments of possibly varying length, where the length is determined by signal bandwidth. The techniques developed use the theory of splines, which give control over smoothness in time and corresponding decay in frequency. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems General method for segmenting Time-Frequency ( R − � R ) space. The idea is to cut up time into segments of possibly varying length, where the length is determined by signal bandwidth. The techniques developed use the theory of splines, which give control over smoothness in time and corresponding decay in frequency. We make our systems so that we have varying degrees of smoothness with cutoffs adaptive to signal bandwidth. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems General method for segmenting Time-Frequency ( R − � R ) space. The idea is to cut up time into segments of possibly varying length, where the length is determined by signal bandwidth. The techniques developed use the theory of splines, which give control over smoothness in time and corresponding decay in frequency. We make our systems so that we have varying degrees of smoothness with cutoffs adaptive to signal bandwidth. We also develop our systems so that the orthogonality of bases in adjacent and possible overlapping blocks is preserved. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Definition (ON Window System) Let 0 < r ≪ T . An ON Window System for adaptive and ultra-wide band sampling is a set of functions { W k ( t ) } such that supp ( W k ( t )) ⊆ [ kT − r , ( k + 1) T + r ] for all k , ( i . ) W k ( t ) ≡ 1 for t ∈ [ kT + r , ( k + 1) T − r ] for all k , ( ii . ) W k (( kT + T / 2) − t ) = W k ( t − ( kT + T / 2)) , t ∈ [0 , T / 2 + r ] , ( iii . ) [ W k ( t )] 2 + [ W k +1 ( t )] 2 = 1 , ( iv . ) ◦ [ n ] } ∈ l 1 . { � W k ( v . ) Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Generate ON Window System by translation of a window W I centered at the origin. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Generate ON Window System by translation of a window W I centered at the origin. Conditions ( i . ) and ( ii . ) are partition properties. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Generate ON Window System by translation of a window W I centered at the origin. Conditions ( i . ) and ( ii . ) are partition properties. Conditions ( iii . ) and ( iv . ) are needed to preserve orthogonality. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Generate ON Window System by translation of a window W I centered at the origin. Conditions ( i . ) and ( ii . ) are partition properties. Conditions ( iii . ) and ( iv . ) are needed to preserve orthogonality. Conditions ( v . ) gives the following. Let f ∈ PW (Ω) and let { W k ( t ) } be a ON Window System with generating window W I . Then � T / 2+ r 1 [ f · W I ] ◦ ( t ) exp( − 2 π int / [ T + 2 r ]) dt T + 2 r − T / 2 − r � f ∗ � W I [ n ] . = Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Examples : { W k ( t ) } = � k ∈ Z χ [( k ) T , ( k +1) T ] ( t ) { W k ( t ) } = � k ∈ Z Cap [( k ) T − r , ( k +1) T + r ] ( t ) , where Cap I ( t ) = 0 | t | ≥ T / 2 + r , 1 | t | ≤ T / 2 − r , sin( π/ (4 r )( t + ( T / 2 + r ))) − T / 2 − r < t < − T / 2 + r , cos( π/ (4 r )( t − ( T / 2 − r ))) T / 2 − r < t < T / 2 + r . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Our general window function W I is k -times differentiable, has supp ( W I ) = [ − T / 2 − r , T / 2 + r ], and has values 0 | t | ≥ T / 2 + r W I = 1 | t | ≤ T / 2 − r ρ ( ± t ) T / 2 − r < | t | < T / 2 + r Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Our general window function W I is k -times differentiable, has supp ( W I ) = [ − T / 2 − r , T / 2 + r ], and has values 0 | t | ≥ T / 2 + r W I = 1 | t | ≤ T / 2 − r ρ ( ± t ) T / 2 − r < | t | < T / 2 + r We solve for ρ ( t ) by solving the Hermite interpolation problem ( a . ) ρ ( T / 2 − r ) = 1 ρ ( n ) ( T / 2 − r ) = 0 , n = 1 , 2 , . . . , k ( b . ) , ρ ( n ) ( T / 2 + r ) = 0 , n = 0 , 2 , . . . , k , ( c . ) [ ρ ( t )] 2 + [ ρ ( − t )] 2 = 1 for t ∈ [ ± ( T / 2 − r ) , ± ( T / 2 + r )] Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Figure: Window W I Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Solving for ρ so that the window in C 1 , we get ρ ( t ) = � � 1 1 − sin( π 2 r ( t + ( T / 2 + r ))) − T / 2 − r < t < − T / 2 , √ 2 �� � � 2 � 1 − 1 sin( π 2 r ( t + ( T / 2 + r ))) − T / 2 < t < − T / 2 + r . 2 Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Adaptive ON Preserving Windowing Systems, Cont’d Solving for ρ so that the window in C 1 , we get ρ ( t ) = � � 1 1 − sin( π 2 r ( t + ( T / 2 + r ))) − T / 2 − r < t < − T / 2 , √ 2 �� � � 2 � 1 − 1 sin( π 2 r ( t + ( T / 2 + r ))) − T / 2 < t < − T / 2 + r . 2 With each degree of smoothness, we get an additional degree of decay in frequency. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality Let { ϕ j ( t ) } be an orthonormal basis for L 2 [ − T / 2 , T / 2]. Define 0 | t | ≥ T / 2 + r ϕ j ( t ) | t | ≤ T / 2 − r ϕ j ( t ) � = − ϕ j ( − T − t ) − T / 2 − r < t < − T / 2 ϕ j ( T − t ) T / 2 < t < T / 2 + r Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Theorem (The Orthogonality of Overlapping Blocks) { Ψ k , j } = { W k � ϕ j ( t ) } is an orthonormal basis for L 2 ( R ) . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Theorem (The Orthogonality of Overlapping Blocks) { Ψ k , j } = { W k � ϕ j ( t ) } is an orthonormal basis for L 2 ( R ) . Sketch of Proof : We want to show that � Ψ k , j , Ψ m , n � = δ k , m · δ j , n . The partitioning properties of the windows give that we need only check overlapping and adjacent windows. Moreover, we need only check window centered at origin. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d � − T / 2 � W I � ϕ i , W I � ( W I ( t )) 2 ϕ i ( − T − t ) ϕ j ( − T − t ) dt ϕ j � = − T / 2 − r � − T / 2+ r (( W I ( t )) 2 − 1) ϕ i ( t ) ϕ j ( t ) dt + − T / 2 � T / 2 + ϕ i ( t ) ϕ j ( t ) dt − T / 2 � T / 2 (( W I ( t )) 2 − 1) ϕ i ( t ) ϕ j ( t ) dt + T / 2 − r � T / 2+ r ( W I ( t )) 2 ϕ i ( T − t ) ϕ j ( T − t ) dt . + T / 2 Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Since { ϕ j } is an ON basis, the third integral equals 1 when i = j . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Since { ϕ j } is an ON basis, the third integral equals 1 when i = j . We apply the linear change of variables t = − T / 2 − τ to the first integral and t = − T / 2 + τ to the second integral. We then add these two integrals together to get � r [( W I ( T / 2 − τ )) 2 +( W I ( τ − T / 2)) 2 − 1] ϕ i ( − T / 2+ τ ) ϕ j ( − T / 2+ τ ) d τ . 0 Conditions ( iii . ) and ( iv . ) give [( W I ( T / 2 − τ )) 2 + ( W I ( τ − T / 2)) 2 − 1] = 0. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Since { ϕ j } is an ON basis, the third integral equals 1 when i = j . We apply the linear change of variables t = − T / 2 − τ to the first integral and t = − T / 2 + τ to the second integral. We then add these two integrals together to get � r [( W I ( T / 2 − τ )) 2 +( W I ( τ − T / 2)) 2 − 1] ϕ i ( − T / 2+ τ ) ϕ j ( − T / 2+ τ ) d τ . 0 Conditions ( iii . ) and ( iv . ) give [( W I ( T / 2 − τ )) 2 + ( W I ( τ − T / 2)) 2 − 1] = 0. Applying the linear change of variables t = T / 2 − τ to the fourth integral and t = T / 2 + τ to the fifth integral gives that these two integrals also sum to zero. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d A similar computation gives that � W k � ϕ i , W k +1 � ϕ j � = 0 . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d A similar computation gives that � W k � ϕ i , W k +1 � ϕ j � = 0 . The partitioning property gives that for | k − l | ≥ 2, � W k � ϕ i , W l � ϕ j � = 0 . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d A similar computation gives that � W k � ϕ i , W k +1 � ϕ j � = 0 . The partitioning property gives that for | k − l | ≥ 2, � W k � ϕ i , W l � ϕ j � = 0 . To finish, we need to show { Ψ k , j } spans L 2 ( R ). Given any function f ∈ L 2 , consider the windowed element f k ( t ) = W k ( t ) · f ( t ). Let f I ( t ) = W I ( t ) · f ( t ). We have that { ϕ j ( t ) } is an orthonormal basis for L 2 [ − T / 2 , T / 2]. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Let f I ( t ) = W I ( t ) · f ( t ). We have that { ϕ j ( t ) } is an orthonormal basis for L 2 [ − T / 2 , T / 2]. Given f I , define ¯ f I ( t ) = 0 | t | ≥ T / 2 + r f I ( t ) | t | ≤ T / 2 − r f I ( t ) − f I ( − T − t ) − T / 2 − r < t < − T / 2 f I ( t ) + f I ( T − t ) T / 2 < t < T / 2 + r Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Since ¯ f I ∈ L 2 [ − T / 2 , T / 2], we may expand it as � ∞ � ¯ � f I , ϕ j ϕ j ( t ) . j =1 Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Since ¯ f I ∈ L 2 [ − T / 2 , T / 2], we may expand it as � ∞ � ¯ � f I , ϕ j ϕ j ( t ) . j =1 To extend this to L 2 [ − T / 2 − r , T / 2 + r ], we expand using { � ϕ j ( t ) } , getting � ∞ � ¯ � � ¯ f I = f I , ϕ j ϕ j ( t ) . � j =1 Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Then � ∞ � ¯ � � ¯ f I = f I , ϕ j ϕ j ( t ) . � j =1 Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d Then � ∞ � ¯ � � ¯ f I = f I , ϕ j ϕ j ( t ) . � j =1 � ¯ f I ( t ) = 0 | t | ≥ T / 2 + r f I ( t ) | t | ≤ T / 2 − r f I ( t ) − f I ( − T − t ) − T / 2 − r < t < − T / 2 + r f I ( t ) + f I ( T − t ) T / 2 − r < t < T / 2 + r This construction preserves orthogonality between adjacent blocks. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory W k Preserve Orthogonality, Cont’d To finish, let f be any function in L 2 . Consider the windowed element f k ( t ) = W k ( t ) · f ( t ). Repeat the construction above for this window. This shows that, for fixed k , { Ψ k , j } spans L 2 ([ kT − r , ( k + 1) T + r ]) and preserves orthogonality between adjacent blocks on either side. Summing over all k ∈ Z gives that { Ψ k , j } is an ON basis for L 2 ( R ). ✷ Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems Similar construction techniques give us partition of unity functions. The theory of B -splines gives us the tools to create these systems. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems Similar construction techniques give us partition of unity functions. The theory of B -splines gives us the tools to create these systems. If we replace condition ( iv . ) with � B k ( t ) ≡ 1 , we get a bounded adaptive partition of unity. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems Similar construction techniques give us partition of unity functions. The theory of B -splines gives us the tools to create these systems. If we replace condition ( iv . ) with � B k ( t ) ≡ 1 , we get a bounded adaptive partition of unity. The systems can be built using B -splines, and have Fourier transforms of the form � sin(2 π T ω ) � n . πω Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Definition (Bounded Adaptive Partition of Unity) A Bounded Adaptive Partition of Unity is a set of functions { B k ( t ) } such that supp ( B k ( t )) ⊆ [ kT − r , ( k + 1) T + r ] , ( i . ) B k ( t ) ≡ 1 for t ∈ [ kT + r , ( k + 1) T − r ] , ( ii . ) B k (( kT + T / 2) − t ) = B k ( t − ( kT + T / 2)) , t ∈ [0 , T / 2 + r ] , ( iii . ) � B k ( t ) ≡ 1 , ( iv . ) k { � ◦ [ n ] } ∈ l 1 . B k ( v . ) Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Conditions ( i . ) , ( ii . ) and ( iv . ) make { B k ( t ) } a bounded partition of unity. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Conditions ( i . ) , ( ii . ) and ( iv . ) make { B k ( t ) } a bounded partition of unity. The change in condition ( iv . ) means that these systems do not preserve orthogonality between blocks. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Conditions ( i . ) , ( ii . ) and ( iv . ) make { B k ( t ) } a bounded partition of unity. The change in condition ( iv . ) means that these systems do not preserve orthogonality between blocks. We will again generate our systems by translations and dilations of a given window B I , where supp ( B I ) = [( − T / 2 − r ) , ( T / 2 + r )]. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Conditions ( i . ) , ( ii . ) and ( iv . ) make { B k ( t ) } a bounded partition of unity. The change in condition ( iv . ) means that these systems do not preserve orthogonality between blocks. We will again generate our systems by translations and dilations of a given window B I , where supp ( B I ) = [( − T / 2 − r ) , ( T / 2 + r )]. Our first example was developed by studying the de la Vall´ ee-Poussin kernel used in Fourier series. Let 0 < r ≪ T and let Tri L ( t ) = max { [((2 T / (4 r )) + r ) − | t | / (2 r )] , 0 } , Tri S ( t ) = max { [((2 T / (4 r )) + r − 1) − | t | / (2 r )] , 0 } and Trap( t ) = Tri L ( t ) − Tri S ( t ) . The Trap function has perfect overlay in the time domain and 1 /ω 2 decay in frequency space. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Examples : { B k ( t ) } = � k ∈ Z χ [( k ) T , ( k +1) T ] ( t ) { B k ( t ) } = � k ∈ Z Trap [( k ) T − r , ( k +1) T + r ] ( t ) . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Examples : { B k ( t ) } = � k ∈ Z χ [( k ) T , ( k +1) T ] ( t ) { B k ( t ) } = � k ∈ Z Trap [( k ) T − r , ( k +1) T + r ] ( t ) . Our general window function W I is k -times differentiable, has supp ( B I ) = [( − T / 2 − r ) , ( T / 2 + r )] and has values 0 | t | ≥ T / 2 + r B I = 1 | t | ≤ T / 2 − r ρ ( ± t ) T / 2 − r < | t | < T / 2 + r Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d We again solve for ρ ( t ) by solving the Hermite interpolation problem ( a . ) ρ ( T / 2 − r ) = 1 ρ ( n ) ( T / 2 − r ) = 0 , n = 1 , 2 , . . . , k ( b . ) ρ ( n ) ( T / 2 + r ) = 0 , n = 0 , 1 , 2 , . . . , k , ( c . ) with the conditions that ρ ∈ C k and [ ρ ( t )] + [ ρ ( − t )] = 1 for t ∈ [ T / 2 − r , T / 2 + r ] . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d We again solve for ρ ( t ) by solving the Hermite interpolation problem ( a . ) ρ ( T / 2 − r ) = 1 ρ ( n ) ( T / 2 − r ) = 0 , n = 1 , 2 , . . . , k ( b . ) ρ ( n ) ( T / 2 + r ) = 0 , n = 0 , 1 , 2 , . . . , k , ( c . ) with the conditions that ρ ∈ C k and [ ρ ( t )] + [ ρ ( − t )] = 1 for t ∈ [ T / 2 − r , T / 2 + r ] . We use B -splines as our cardinal functions. Let 0 < α ≪ β and consider χ [ − α,α ] . We want the n -fold convolution of χ [ α,α ] to fit in the interval [ − β, β ]. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Then we choose α so that 0 < n α < β and let Ψ( t ) = χ [ − α,α ] ∗ χ [ − α,α ] ∗ · · · ∗ χ [ − α,α ] ( t ) . � �� � n − times Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d Then we choose α so that 0 < n α < β and let Ψ( t ) = χ [ − α,α ] ∗ χ [ − α,α ] ∗ · · · ∗ χ [ − α,α ] ( t ) . � �� � n − times The β -periodic continuation of this function, Ψ ◦ ( t ) has the Fourier series expansion � sin( π k α/ n β ) � n � α exp( π ikt /β ) . n β 2 π k α/ n β k � =0 Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d The C k solution for ρ is given by a theorem of Schoenberg. Schoenberg solved the Hermite interpolation problem S ( n ) ( − 1) = 0 , n = 0 , 1 , 2 , . . . , k , ( a . ) ( b . ) S (1) = 1 , S ( n ) (1) = 0 , n = 1 , 2 , . . . , k . ( b . ) Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d The C k solution for ρ is given by a theorem of Schoenberg. Schoenberg solved the Hermite interpolation problem S ( n ) ( − 1) = 0 , n = 0 , 1 , 2 , . . . , k , ( a . ) ( b . ) S (1) = 1 , S ( n ) (1) = 0 , n = 1 , 2 , . . . , k . ( b . ) An interpolant that minimizes the Chebyshev norm is called the perfect spline . The perfect spline S ( t ) for Hermite problem above is given by the integral of the function k � Ψ( t − t j ) M ( x ) = ( − 1) n , φ ′ ( t j ) j =0 where Ψ is the ( k + 1) convolution of characteristic functions, the k ) and φ ( t ) = � k knot points are t j = − cos( π j j =0 ( t − t j ). Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d We then have that ρ ( t ) = S ◦ ℓ ( t ) , where ℓ ( t ) = 1 r t − 2 T 2 r . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Partition of Unity Systems, Cont’d We then have that ρ ( t ) = S ◦ ℓ ( t ) , where ℓ ( t ) = 1 r t − 2 T 2 r . For this ρ , and for 0 | t | ≥ T / 2 + r B I = 1 | t | ≤ T / 2 − r ρ ( ± t ) T / 2 − r < | t | < T / 2 + r we have that � B I ( ω ) is given by the antiderivative of a linear combination of functions of the form � sin(2 π T ω ) � k +1 , πω and therefore has decay 1 /ω k +2 in frequency. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Almost ON Systems Cotlar, Knapp and Stein introduced almost orthogonality via operator inequalities. Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Almost ON Systems Cotlar, Knapp and Stein introduced almost orthogonality via operator inequalities. We are looking to create windowing systems that are more computable/constructible such as the Bounded Adaptive Partition of Unity systems { B k ( t ) } with the orthogonality preservation of the ON Window System { W k ( t ) } . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Almost ON Systems Cotlar, Knapp and Stein introduced almost orthogonality via operator inequalities. We are looking to create windowing systems that are more computable/constructible such as the Bounded Adaptive Partition of Unity systems { B k ( t ) } with the orthogonality preservation of the ON Window System { W k ( t ) } . Consider { W k ( t ) } = � k ∈ Z Cap [( k ) T − r , ( k +1) T + r ] ( t ) , where Cap I ( t ) = 0 | t | ≥ T / 2 + r , 1 | t | ≤ T / 2 − r , sin( π/ (4 r )( t + ( T / 2 + r ))) − T / 2 − r < t < − T / 2 + r , cos( π/ (4 r )( t − ( T / 2 − r ))) T / 2 − r < t < T / 2 + r . Stephen Casey Signal Adaptive Frame Theory
Preliminary Definitions W-K-S Sampling Projection Method ON Window Systems Adaptive Windowing Systems Partition of Unity Systems Projection Revisited Almost ON Systems Binary Signals and Walsh Functions Signal Adaptive Frame Theory Almost ON Systems, Cont’d Definition (Almost ON System) Let 0 < r ≪ T . An Almost ON System for adaptive and ultra-wide band sampling is a set of functions { A k ( t ) } for which there exists δ , 0 ≤ δ ≤ 1 / 2, such that supp ( A k ( t )) ⊆ [ kT − r , ( k + 1) T + r ] for all k , ( i . ) A k ( t ) ≡ 1 for t ∈ [ kT + r , ( k + 1) T − r ] for all k , ( ii . ) A k (( kT + T / 2) − t ) = A k ( t − ( kT + T / 2)) , t ∈ [0 , T / 2 + r ] , ( iii . ) 1 − δ ≤ [ A k ( t )] 2 + [ A k +1 ( t )] 2 ≤ 1 + δ , ( iv . ) ◦ [ n ] } ∈ l 1 . { � A k ( v . ) Stephen Casey Signal Adaptive Frame Theory
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