Signal decomposition via SuperEMD Maria van der Walt 1 Joint work - PowerPoint PPT Presentation

EMD + Hilbert spectral analysis Limitations of EMD with Hilbert spectral analysis: • The sifting process cannot distinguish between atoms with frequencies that are very close together. In other words, a single IMF may actually contain more than one atom. • We can get inaccurate results when computing the Hilbert transform from discrete samples on a bounded interval. • The standard cubic spline interpolation scheme used to construct the upper and lower envelopes in the sifting process is not a local method. Maria van der Walt | Westmont College 8/ 21

Our contribution Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j ... Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j ... Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j ... • non-decreasing knot sequence t • m th order B-splines: � 1 , if t j ≤ t < t j +1 N t , 1 ,j ( t ) := 0 , otherwise N t ,m,j ( t ) := w m,j N t ,m − 1 ,j ( t ) + (1 − w m,j +1 ) N t ,m − 1 ,j +1 ( t ) t − t j � if t j � = t j + m − 1 t j + m − 1 − t j with w m,j ( t ) := 0 , otherwise Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j ... Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: (i) P has a local formulation; Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: (i) P has a local formulation; (ii) P preserves polynomials of degree ≤ 3 ; Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: (i) P has a local formulation; (ii) P preserves polynomials of degree ≤ 3 ; (iii) P f interpolates f at a = t 0 < t 1 < · · · < t n = b ; Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: (i) P has a local formulation; (ii) P preserves polynomials of degree ≤ 3 ; (iii) P f interpolates f at a = t 0 < t 1 < · · · < t n = b ; (iv) P preserves derivatives (up to order 3) of f at a and b . Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: (i) P has a local formulation; Q (ii) P preserves polynomials of degree ≤ 3 ; Q (iii) P f interpolates f at a = t 0 < t 1 < · · · < t n = b ; (iv) P preserves derivatives (up to order 3) of f at a and b . Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: (i) P has a local formulation; Q , R (ii) P preserves polynomials of degree ≤ 3 ; Q (iii) P f interpolates f at a = t 0 < t 1 < · · · < t n = b ; R (iv) P preserves derivatives (up to order 3) of f at a and b . R Maria van der Walt | Westmont College 9/ 21

Our contribution Our solution to third limitation: Given a function f , we construct an interpolation operator P in terms of cubic B-splines N t , 4 ,j such that: (i) P has a local formulation; Q , R (ii) P preserves polynomials of degree ≤ 3 ; Q (iii) P f interpolates f at a = t 0 < t 1 < · · · < t n = b ; R (iv) P preserves derivatives (up to order 3) of f at a and b . R Blending operator [Chui, Diamond, 1990]: P := Q + R − RQ Maria van der Walt | Westmont College 9/ 21

Blending operator Quasi-interpolant ( Q f )( t ) := − 1 n n +2 � � � f ( − j ) ( a ) M j ( t ) + f ( j − n ) ( b ) M j ( t ) f ( t j ) M j ( t ) + j = − 3 j =0 j = n +1 Maria van der Walt | Westmont College 10/ 21

Blending operator Quasi-interpolant ( Q f )( t ) := − 1 n n +2 � � � f ( − j ) ( a ) M j ( t ) + f ( j − n ) ( b ) M j ( t ) f ( t j ) M j ( t ) + j = − 3 j =0 j = n +1 Spline molecules [Chen, Chui, Lai, 1988; Chui, vdW, 2015]:  j +3 �  a j,k N t , 4 ,k − 3 ( t ) , j = − 3 , . . . , − 1;      k =0     3   � a j,k N t , 4 ,j + k − 3 ( t ) , j = 0 , . . . , n − 1; M j ( t ) =  k =0    2    �  a j,k N t , 4 ,n + k − 3 ( t ) , j = n, . . . , n + 2 .     k = j − n Maria van der Walt | Westmont College 10/ 21

Blending operator Quasi-interpolant ( Q f )( t ) := − 1 n n +2 � � � f ( − j ) ( a ) M j ( t ) + f ( j − n ) ( b ) M j ( t ) f ( t j ) M j ( t ) + j = − 3 j =0 j = n +1 Spline molecules [Chen, Chui, Lai, 1988; Chui, vdW, 2015]:  j +3 �  a j,k N t , 4 ,k − 3 ( t ) , j = − 3 , . . . , − 1;      k =0     3   � a j,k N t , 4 ,j + k − 3 ( t ) , j = 0 , . . . , n − 1; M j ( t ) =  k =0    2    �  a j,k N t , 4 ,n + k − 3 ( t ) , j = n, . . . , n + 2 .     k = j − n Find spline coefficients a j,k such that ( Q p )( t ) = p ( t ) , p ∈ π 3 . Maria van der Walt | Westmont College 10/ 21

Blending operator Local interpolant ( R f )( t ) := − 1 n n +2 � f ( − j ) ( a ) L j ( t ) + � � f ( j − n ) ( b ) L j ( t ) f ( t j ) L j ( t ) + j = − 3 j =0 j = n +1 Maria van der Walt | Westmont College 11/ 21

Blending operator Local interpolant ( R f )( t ) := − 1 n n +2 � f ( − j ) ( a ) L j ( t ) + � � f ( j − n ) ( b ) L j ( t ) f ( t j ) L j ( t ) + j = − 3 j =0 j = n +1 Spline molecules [Chui, vdW, 2015]:  3 �  b j,k N ˜ t k − 3 , 4 ( t ) , j = − 3 , . . . , 0;      k =0    t , 4 , 2 j ( t )  N ˜  L j ( t ) = j = 1 , . . . , n − 1; t , 4 , 2 j ( t j ) , N ˜     2   �  t n + k , 4 ( t ) , j = n, . . . , n + 2 . b j,k N ˜     k =0 Maria van der Walt | Westmont College 11/ 21

Blending operator Local interpolant ( R f )( t ) := − 1 n n +2 � f ( − j ) ( a ) L j ( t ) + � � f ( j − n ) ( b ) L j ( t ) f ( t j ) L j ( t ) + j = − 3 j =0 j = n +1 Spline molecules [Chui, vdW, 2015]: Maria van der Walt | Westmont College 11/ 21

Blending operator Local interpolant ( R f )( t ) := − 1 n n +2 � f ( − j ) ( a ) L j ( t ) + � � f ( j − n ) ( b ) L j ( t ) f ( t j ) L j ( t ) + j = − 3 j =0 j = n +1 Spline molecules [Chui, vdW, 2015]:  3 �  b j,k N ˜ t k − 3 , 4 ( t ) , j = − 3 , . . . , 0;      k =0    t , 4 , 2 j ( t )  N ˜  L j ( t ) = j = 1 , . . . , n − 1; → L j ( t ℓ ) = δ j,ℓ t , 4 , 2 j ( t j ) , N ˜     2   �  t n + k , 4 ( t ) , j = n, . . . , n + 2 . b j,k N ˜     k =0 Maria van der Walt | Westmont College 11/ 21

Blending operator Local interpolant ( R f )( t ) := − 1 n n +2 � f ( − j ) ( a ) L j ( t ) + � � f ( j − n ) ( b ) L j ( t ) f ( t j ) L j ( t ) + j = − 3 j =0 j = n +1 Spline molecules [Chui, vdW, 2015]:  3 → L ( ℓ ) �  b j,k N ˜ t k − 3 , 4 ( t ) , j = − 3 , . . . , 0; j ( a ) = δ − j,ℓ      k =0    t , 4 , 2 j ( t )  N ˜  L j ( t ) = j = 1 , . . . , n − 1; → L j ( t ℓ ) = δ j,ℓ t , 4 , 2 j ( t j ) , N ˜     2   → L ( ℓ ) �  t n + k , 4 ( t ) , j = n, . . . , n + 2 . j ( b ) = δ j − n,ℓ b j,k N ˜     k =0 Maria van der Walt | Westmont College 11/ 21

Blending operator Local interpolant ( R f )( t ) := − 1 n n +2 � f ( − j ) ( a ) L j ( t ) + � � f ( j − n ) ( b ) L j ( t ) f ( t j ) L j ( t ) + j = − 3 j =0 j = n +1 Spline molecules [Chui, vdW, 2015]:  3 → L ( ℓ ) �  b j,k N ˜ t k − 3 , 4 ( t ) , j = − 3 , . . . , 0; j ( a ) = δ − j,ℓ      k =0    t , 4 , 2 j ( t )  N ˜  L j ( t ) = j = 1 , . . . , n − 1; → L j ( t ℓ ) = δ j,ℓ t , 4 , 2 j ( t j ) , N ˜     2   → L ( ℓ ) �  t n + k , 4 ( t ) , j = n, . . . , n + 2 . j ( b ) = δ j − n,ℓ b j,k N ˜     k =0 Easy to show: ( R f )( t ℓ )= f ( t ℓ ) and R preserves derivatives at a, b . Maria van der Walt | Westmont College 11/ 21

Blending operator Blending: P := Q + R − RQ Therefore [Chui, vdW, 2015]: Maria van der Walt | Westmont College 12/ 21

Blending operator Blending: P := Q + R − RQ Therefore [Chui, vdW, 2015]: ( P p )( t ) = ( Q p )( t ) + ( R p )( t ) − ( R ( Q p ))( t ) = p ( t ) + ( R p )( t ) − ( R p )( t ) = p ( t ) , p ∈ π 3 ; Maria van der Walt | Westmont College 12/ 21

Blending operator Blending: P := Q + R − RQ Therefore [Chui, vdW, 2015]: ( P p )( t ) = p ( t ) , p ∈ π 3 ; ( P f )( t j ) = ( Q f )( t j ) + ( R f )( t j ) − ( R ( Q f ))( t j ) = ( Q f )( t j ) + f ( t j ) − ( Q f )( t j ) = f ( t j ) , j = 0 , . . . , n ; Maria van der Walt | Westmont College 12/ 21

Blending operator Blending: P := Q + R − RQ Therefore [Chui, vdW, 2015]: ( P p )( t ) = p ( t ) , p ∈ π 3 ; ( P f )( t j ) = f ( t j ) , j = 0 , . . . , n ; ( P f ) ( j ) ( a ) = f ( j ) ( a ) , j = 1 , 2 , 3; ( P f ) ( j ) ( b ) = f ( j ) ( b ) , j = 1 , 2 . Maria van der Walt | Westmont College 12/ 21

EMD + Hilbert spectral analysis Limitations of EMD with Hilbert spectral analysis: • The sifting process cannot distinguish between atoms with frequencies that are very close together. In other words, a single IMF may actually contain more than one atom. • We can get inaccurate results when computing the Hilbert transform from discrete samples on a bounded interval. • The standard cubic spline interpolation scheme used to construct the upper and lower envelopes in the sifting process is not a local method. Maria van der Walt | Westmont College 13/ 21

EMD + Hilbert spectral analysis Limitations of EMD with Hilbert spectral analysis: • The sifting process cannot distinguish between atoms with frequencies that are very close together. In other words, a single IMF may actually contain more than one atom. • We can get inaccurate results when computing the Hilbert transform from discrete samples on a bounded interval. � The standard cubic spline interpolation scheme used to construct the upper and lower envelopes in the sifting process is not a local method – use blending operator. Maria van der Walt | Westmont College 13/ 21

EMD + Hilbert spectral analysis Limitations of EMD with Hilbert spectral analysis: • The sifting process cannot distinguish between atoms with frequencies that are very close together . In other words, a single IMF may actually contain more than one atom. • We can get inaccurate results when computing the Hilbert transform from discrete samples on a bounded interval. � The standard cubic spline interpolation scheme used to construct the upper and lower envelopes in the sifting process is not a local method – use blending operator. Maria van der Walt | Westmont College 13/ 21

Our contribution Our solution to first two limitations: Maria van der Walt | Westmont College 14/ 21

Our contribution Our solution to first two limitations: • Apply signal separation operator (SSO) [Chui, Mhaskar, 2015] to each IMF from EMD separately (instead of Hilbert spectral analysis). Maria van der Walt | Westmont College 14/ 21

Our contribution Our solution to first two limitations: • Apply signal separation operator (SSO) [Chui, Mhaskar, 2015] to each IMF from EMD separately (instead of Hilbert spectral analysis). • SSO is able to identify very close-by frequencies in a given IMF and recover/reconstruct the individual atoms associated with these frequencies. Maria van der Walt | Westmont College 14/ 21

Our contribution Our solution to first two limitations: • Apply signal separation operator (SSO) [Chui, Mhaskar, 2015] to each IMF from EMD separately (instead of Hilbert spectral analysis). • SSO is able to identify very close-by frequencies in a given IMF and recover/reconstruct the individual atoms associated with these frequencies. • SSO is a direct, local method which produces more accurate results in near real-time. Maria van der Walt | Westmont College 14/ 21

Our contribution Our solution to first two limitations: • Apply signal separation operator (SSO) [Chui, Mhaskar, 2015] to each IMF from EMD separately (instead of Hilbert spectral analysis). • SSO is able to identify very close-by frequencies in a given IMF and recover/reconstruct the individual atoms associated with these frequencies. • SSO is a direct, local method which produces more accurate results in near real-time. • EMD + SSO = “SuperEMD” Maria van der Walt | Westmont College 14/ 21

AHM Adaptive harmonic model (AHM): N � f ( t ) = A 0 ( t ) + A j ( t ) cos(2 πφ j ( t )) , t ∈ R j =1 Maria van der Walt | Westmont College 15/ 21

AHM Adaptive harmonic model (AHM): N � f ( t ) = A 0 ( t ) + A j ( t ) cos(2 πφ j ( t )) , t ∈ R j =1 Assumptions: • A j ∈ C ( R ) , A j ( t ) > 0 and φ j ∈ C 1 ( R ) , φ ′ j ( t ) > 0 • There exists α = α ( t ) > 0 s.t. for any u with | u | ≤ α − 1 (8 πB ) − 1 / 2 , where B = B ( t ) := max φ ′ j ( t ) : j • | A j ( t + u ) − A j ( t ) | ≤ α 3 | u | A j ( t ) • | φ ′ j ( t + u ) − φ ′ j ( t ) | ≤ α 3 | u | φ ′ j ( t ) k ( t ) | =: 2 Bη j � = k | φ ′ j ( t ) − φ ′ • There exists η = η ( t ) s.t. min > 0 π Note: N � M = M ( t ) := A j ( t ) and µ = µ ( t ) := min 1 ≤ j ≤ N A j ( t ) j =1 Maria van der Walt | Westmont College 15/ 21

SuperEMD Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: � ℓ � 1 � e iℓθ f j ( t − ℓδ ) ( T a,δ f j )( t, θ ) = h � k � � a k ∈ Z h a ℓ ∈ Z ( t ∈ R , θ ∈ [ − π, π ]) Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: � ℓ � 1 � e iℓθ f j ( t − ℓδ ) ( T a,δ f j )( t, θ ) = h � k � � a k ∈ Z h a ℓ ∈ Z ( t ∈ R , θ ∈ [ − π, π ]) • h = window function (non-neg, even, supported on [ − 1 , 1]) Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: � ℓ � 1 � e iℓθ f j ( t − ℓδ ) ( T a,δ f j )( t, θ ) = h � k � � a k ∈ Z h a ℓ ∈ Z ( t ∈ R , θ ∈ [ − π, π ]) • h = window function (non-neg, even, supported on [ − 1 , 1]) � k � • a = window width (chosen so that � k ∈ Z h > 0) a Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: � ℓ � 1 � e iℓθ f j ( t − ℓδ ) ( T a,δ f j )( t, θ ) = h � k � � a k ∈ Z h a ℓ ∈ Z ( t ∈ R , θ ∈ [ − π, π ]) • h = window function (non-neg, even, supported on [ − 1 , 1]) � k � • a = window width (chosen so that � k ∈ Z h > 0) a • δ = sample spacing (adjust based on separation of IF’s) Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: � ℓ � 1 � e iℓθ f j ( t − ℓδ ) ( T a,δ f j )( t, θ ) = h � k � � a k ∈ Z h a ℓ ∈ Z ( t ∈ R , θ ∈ [ − π, π ]) Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 � � • Φ a h ; θ − 2 πδφ ′ j,n ( t ) ≈ Dirac delta function at 0 [Mhaskar, Prestin, 2000] Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 � � • Φ a h ; θ − 2 πδφ ′ j,n ( t ) ≈ Dirac delta function at 0 [Mhaskar, Prestin, 2000] • . . . → true atom f j,n ( t ) = A j,n ( t ) cos(2 πφ j,n ( t )) Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 • It can be shown [Chui, Mhaskar 2015] that θ ∈ [0 , π ] : | ( T a,δ f j ) ( t, θ ) | ≥ µ � � 2 consists of disjoint clusters G j, 1 , . . . , G j,N j , centered around 2 πδφ ′ j, 1 ( t ) , . . . , 2 πδφ ′ j,N j ( t ) . Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 • Compute true IF’s: 1 φ ′ j,n ( t ) ≈ 2 πδ arg max θ ∈ G j,n | ( T a,δ f j ) ( t, θ ) | Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 • Compute true IF’s: 1 φ ′ j,n ( t ) ≈ 2 πδ arg max θ ∈ G j,n | ( T a,δ f j ) ( t, θ ) | • Recover true atoms: f j,n ( t ) ≈ 2 Re ( T a,δ f j ) ( t, 2 πδφ ′ j,n ( t )) Maria van der Walt | Westmont College 16/ 21

SuperEMD Given IMF f j ( t ) from (modified) EMD, containing N j true atoms. • Apply SSO to IMF f j [Chui, Mhaskar, vdW 2016]: N j � A j,n ( t ) e 2 πiφ j,n ( t ) Φ a h ; θ − 2 πδφ ′ � � ( T a,δ f j )( t, θ ) ≈ j,n ( t ) n =1 • Compute true IF’s: � � 1 � � φ ′ � j,n ( t ) − 2 πδ arg max θ ∈ G j,n | ( T a,δ f j ) ( t, θ ) | � ≤ K 1 ( α, B, M, µ, h ) � � • Recover true atoms: � � � f j,n ( t ) − 2 Re ( T a,δ f j ) ( t, 2 πδφ ′ j,n ( t )) � ≤ K 2 ( α, M, µ ) � � √ � � 0 < δ ≤ (4 B ) − 1 and a = ( αδ 8 πB ) − 1 for sufficiently small α Maria van der Walt | Westmont College 16/ 21

Examples Maria van der Walt | Westmont College 17/ 21

Examples f ( t ) = cos 2 π (5 t ) + cos 2 π (4 . 9 t ) Maria van der Walt | Westmont College 17/ 21

Examples ( a = 1800 , δ = 1 f ( t ) = cos 2 π (5 t ) + cos 2 π (4 . 9 t ) 20 ) Maria van der Walt | Westmont College 17/ 21

Examples ( a = 1800 , δ = 1 f ( t ) = cos 2 π (5 t ) + cos 2 π (4 . 9 t ) 20 ) Maria van der Walt | Westmont College 17/ 21

Examples ( a = 1800 , δ = 1 f ( t ) = cos 2 π (5 t ) + cos 2 π (4 . 9 t ) + noise 20 ) Maria van der Walt | Westmont College 17/ 21