Performance Analysis and Optimal Filter Design for Sigma-Delta Modulation via Duality with DPCM Or Ordentlich Joint work with Uri Erez ISIT 2015, Hong Kong June 15, 2015 Ordentlich and Erez Sigma-Delta/DPCM Duality
Oversampled Data Conversion X ( t ) is a stationary Gaussian process with S X ( f ) = 0, ∀| f | > f max Sampling X ( t ) at Nyquist’s rate gives the discrete process X n Sampling X ( t ) at L x Nyquist’s rate gives the discrete process X L n Ordentlich and Erez Sigma-Delta/DPCM Duality
Oversampled Data Conversion X ( t ) is a stationary Gaussian process with S X ( f ) = 0, ∀| f | > f max Sampling X ( t ) at Nyquist’s rate gives the discrete process X n Sampling X ( t ) at L x Nyquist’s rate gives the discrete process X L n Rate-Distortion 101 The number of bits per second for describing both processes with distortion D is equal Normalizing by the number of samples per second gives R X L ( D ) = 1 L · R X ( D ) Ordentlich and Erez Sigma-Delta/DPCM Duality
Oversampled Data Conversion X ( t ) is a stationary Gaussian process with S X ( f ) = 0, ∀| f | > f max Sampling X ( t ) at Nyquist’s rate gives the discrete process X n Sampling X ( t ) at L x Nyquist’s rate gives the discrete process X L n Rate-Distortion 101 The number of bits per second for describing both processes with distortion D is equal Normalizing by the number of samples per second gives R X L ( D ) = 1 L · R X ( D ) In data conversion fast low-resolution ADCs are often preferable over slow high-resolution ADCs Ordentlich and Erez Sigma-Delta/DPCM Duality
Σ∆ Modulation Standard Data Conversion X n ˆ X ( t ) Sampler Q ( · ) X n T s = 1 / 2 f max Ordentlich and Erez Sigma-Delta/DPCM Duality
Σ∆ Modulation Standard Data Conversion X n ˆ X ( t ) Sampler Q ( · ) X n T s = 1 / 2 f max Oversampled Data Conversion H ( ω ) 1 X L ˆ X ( t ) Sampler n Q ( · ) X n ω − π π L L T s = 1 / 2 Lf max Ordentlich and Erez Sigma-Delta/DPCM Duality
Σ∆ Modulation Standard Data Conversion X n ˆ X ( t ) Sampler Q ( · ) X n T s = 1 / 2 f max Oversampled Data Conversion H ( ω ) 1 X L ˆ X ( t ) Sampler n Q ( · ) X n ω − π π L L T s = 1 / 2 Lf max Oversampling reduces the MSE distortion by 1 / L ⇒ Not good enough, want exponential decay with L Ordentlich and Erez Sigma-Delta/DPCM Duality
Σ∆ Modulation Standard Data Conversion X n ˆ X ( t ) Sampler Q ( · ) X n T s = 1 / 2 f max Σ∆ Modulation H ( ω ) ˆ 1 U n U n ˆ X L X L Q ( · ) Σ n n − − π π ω L L − C ( Z ) Σ N n Ordentlich and Erez Sigma-Delta/DPCM Duality
Σ∆ Modulation Standard Data Conversion X n ˆ X ( t ) Sampler Q ( · ) X n T s = 1 / 2 f max Σ∆ Modulation H ( ω ) ˆ 1 U n U n ˆ X L X L Q ( · ) Σ n n − − π π ω L L − C ( Z ) Σ N n Our goal is to analyze the performance of Σ∆: Quantization rate vs. MSE distortion Ordentlich and Erez Sigma-Delta/DPCM Duality
Σ∆ Modulation Standard Data Conversion X n ˆ X ( t ) Sampler Q ( · ) X n T s = 1 / 2 f max Σ∆ Modulation H ( ω ) ˆ 1 U n U n ˆ X L X L Q ( · ) Σ n n − − π π ω L L − C ( Z ) Σ N n We will model the Σ∆ modulator by a test-channel Ordentlich and Erez Sigma-Delta/DPCM Duality
Σ∆ Modulation Standard Data Conversion X n ˆ X ( t ) Sampler Q ( · ) X n T s = 1 / 2 f max Σ∆ Modulation 0 , σ 2 � � N n ∼ N Σ∆ H ( ω ) 1 U n U n + N n ˆ X L X L Σ n n − − π π ω L L − C ( Z ) Σ N n Will study the tradeoff between I ( U n ; U n + N n ) and the MSE distortion E ( ˆ n ) 2 X L n − X L Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization ˆ X n Q ( · ) X n Q ( x ): · · · 1 2 3 2 R x √ √ 12 σ 2 12 σ 2 Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization ˆ X n Q ( · ) X n Q ( x ): · · · 1 2 3 2 R x OVERLOAD OVERLOAD √ √ 12 σ 2 12 σ 2 No Overload Region Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization ˆ X n Q ( · ) X n Q ( x ): · · · 1 2 3 2 R x OVERLOAD OVERLOAD √ √ 12 σ 2 12 σ 2 No Overload Region High-resolution/dithered quantization assumption + no overload √ √ � � ˆ 12 σ 2 12 σ 2 N n ∼ Uniform − X n = X n + N n ; , , X n = N n 2 2 | Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization ˆ X n Q ( · ) X n Q ( x ): · · · 1 2 3 2 R x OVERLOAD OVERLOAD √ √ 12 σ 2 12 σ 2 No Overload Region | X n + N n | < 2 R √ 12 σ 2 2 High-resolution/dithered quantization assumption + no overload √ √ � � ˆ 12 σ 2 12 σ 2 N n ∼ Uniform − X n = X n + N n ; , , X n = N n 2 2 | Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization � 2 � ˆ ˆ = σ 2 X n X n − X n X n E √ √ � � 12 σ 2 12 σ 2 N n ∼ Uniform − , 2 2 Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization � 2 � ˆ ˆ = σ 2 X n X n − X n X n E √ √ � � 12 σ 2 12 σ 2 N n ∼ Uniform − , 2 2 Recalling X n ∼ N (0 , σ 2 X ), it is easy to show | X n + N n | > 2 R √ σ 2 � � �� � � � � 12 σ 2 R − 1 − 3 X 2 2 log 1+ σ 2 P ol � Pr ≤ 2exp 22 2 Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization � 2 � ˆ ˆ = σ 2 X n X n − X n X n E N n ∼ N (0 , σ 2 ) Recalling X n ∼ N (0 , σ 2 X ), it is easy to show � � − 3 22 2( R − I ( X n ; X n + N n )) P ol ≤ 2exp Ordentlich and Erez Sigma-Delta/DPCM Duality
Relevance of Gaussian Test Channel Uniform Scalar Quantization � 2 � ˆ ˆ = σ 2 X n X n − X n X n E N n ∼ N (0 , σ 2 ) Recalling X n ∼ N (0 , σ 2 X ), it is easy to show � � − 3 22 2( R − I ( X n ; X n + N n )) P ol ≤ 2exp Conclusion: the quantizer can be replaced by an AWGN test-channel Ordentlich and Erez Sigma-Delta/DPCM Duality
Back to the Σ∆ Test Channel � 0 , σ 2 � N n ∼ N Σ∆ H ( ω ) 1 U n U n + N n ˆ X Σ∆ X Σ∆ Σ n − n − π π ω L L − C ( Z ) Σ N n Ordentlich and Erez Sigma-Delta/DPCM Duality
Back to the Σ∆ Test Channel � 0 , σ 2 � N n ∼ N Σ∆ H ( ω ) 1 U n U n + N n ˆ X Σ∆ X Σ∆ Σ n − n − π π ω L L − C ( Z ) Σ N n U n = X Σ∆ − c n ∗ N n n U n + N n = X Σ∆ + ( δ n − c n ) ∗ N n n � 1 + E ( U n ) 2 � I ( U n ; U n + N n ) = 1 2 log σ 2 Σ∆ ˆ X n = X Σ∆ + h n ∗ ( δ n − c n ) ∗ N n n − ˆ X Σ∆ X Σ∆ = h n ∗ ( δ n − c n ) ∗ N n n n Ordentlich and Erez Sigma-Delta/DPCM Duality
Back to the Σ∆ Test Channel 0 , σ 2 � � N n ∼ N Σ∆ H ( ω ) 1 U n U n + N n X Σ∆ ˆ X Σ∆ Σ n − n ω − π π L L − C ( Z ) Σ N n Proposition - Σ∆ Rate-Distortion Tradeoff For any stationary Gaussian process with variance σ 2 X sampled L times above Nyquist’s rate � π | C ( ω ) | 2 d ω + σ 2 I ( U n ; U n + N n ) = 1 � 1 + 1 � X 2 log , σ 2 2 π − π Σ∆ � π/ L Σ∆ · 1 D = σ 2 | 1 − C ( ω ) | 2 d ω 2 π − π/ L Ordentlich and Erez Sigma-Delta/DPCM Duality
Back to the Σ∆ Test Channel � 0 , σ 2 � N n ∼ N Σ∆ H ( ω ) 1 U n U n + N n ˆ X Σ∆ X Σ∆ Σ n − n − π π ω L L − C ( Z ) Σ N n Not clear how to choose C ( Z ) Ordentlich and Erez Sigma-Delta/DPCM Duality
Detour: DPCM � 0 , σ 2 � N n ∼ N DPCM H ( ω ) 1 U n U n + N n Σ V n X DPCM ˆ X DPCM Σ n − n + − π π ω L L C ( Z ) Ordentlich and Erez Sigma-Delta/DPCM Duality
Detour: DPCM � 0 , σ 2 � N n ∼ N DPCM H ( ω ) 1 U n U n + N n Σ V n X DPCM ˆ X DPCM Σ n − n + − π π ω L L C ( Z ) Popular for compression of stationary processes (rather than A/D) Design depends on 2 nd -order statistics of { X DPCM } (in contrast to Σ∆) n Rate-Distortion tradeoff of DPCM is well understood (McDonald66, JN84, ZKE08) Ordentlich and Erez Sigma-Delta/DPCM Duality
Detour: DPCM � 0 , σ 2 � N n ∼ N DPCM H ( ω ) 1 U n U n + N n Σ V n X DPCM ˆ X DPCM Σ n n − + − π π ω L L C ( Z ) DPCM Rate-Distortion Tradeoff for Flat Low-Pass Process Let { X DPCM } be a stationary Gaussian process with PSD n � L σ 2 for | ω | ≤ π/ L S DPCM X ( ω ) = for π/ L < | ω | < π , X 0 then D = σ 2 DPCM / L and � π � π/ L | C ( ω ) | 2 d ω + L σ 2 � � I ( U n ; U n + N n ) = 1 1+ 1 1 | 1 − C ( ω ) | 2 d ω X 2 log σ 2 2 π 2 π − π DPCM − π/ L Ordentlich and Erez Sigma-Delta/DPCM Duality
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