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ESE 531: Digital Signal Processing

Lec 10: February 14th, 2017 Practical and Non-integer Sampling, Multi-

rate Sampling

Penn ESE 531 Spring 2017 - Khanna

Lecture Outline

!  Downsampling/Upsampling !  Practical Interpolation !  Non-integer Resampling !  Multi-Rate Processing

"  Interchanging Operations

!  Polyphase Decomposition !  Multi-Rate Filter Banks

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Downsampling

!  Definition: Reducing the sampling rate by an integer number

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Downsampling

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Example

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4π 2π

Example

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6π 4π 2π

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Example

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Example

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Upsampling

!  Definition: Increasing the sampling rate by an integer number

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x[n]= xc (nT )xi[n]= xc (nT ')

Upsampling

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xi[n]

Frequency Domain Interpretation

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Frequency Domain Interpretation

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Example

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Example

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Example

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Example

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Example

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Example

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Practical Interpolation

!  Interpolate with simple, practical filters "  Linear interpolation – samples between original samples

fall on a straight line connecting the samples "  Convolve with triangle instead of sinc

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Practical Interpolation

!  Interpolate with simple, practical filters "  Linear interpolation – samples between original samples

fall on a straight line connecting the samples "  Convolve with triangle instead of sinc

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Frequency Domain Interpretation

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Linear Interpolation -- Frequency Domain

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xi[n]= xe[n]∗hlin[n]

LPF approx

Linear Interpolation -- Frequency Domain

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xi[n]= xe[n]∗hlin[n]

LPF approx

Linear Interpolation -- Frequency Domain

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xi[n]= xe[n]∗hlin[n]

LPF approx

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Non-integer Sampling

!  T’=TM/L "  Upsample by L, then downsample by M

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interpolator decimator

Non-integer Sampling

!  T’=TM/L "  Upsample by L, then downsample by M

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interpolator decimator

Example

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!  T’=3/2T # L=2, M=3

Example

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!  T’=3/2T # L=2, M=3

Non-integer Sampling

!  T’=TM/L "  Downsample by M, then upsample by L?

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interpolator decimator

Multi-Rate Signal Processing

!  What if we want to resample by 1.01T? "  Expand by L=100 "  Filter π/101 ($$$$$) "  Downsample by M=101

!  Fortunately there are ways around it! "  Called multi-rate "  Uses compressors, expanders and filtering

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Interchanging Operations

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Upsampling -expanding in time -compressing in frequency

Downsampling -compressing in time -expanding in frequency

Interchanging Operations - Expander

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Upsampling -expanding in time -compressing in frequency

?

Interchanging Operations - Expander

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Upsampling -expanding in time -compressing in frequency

?

Interchanging Operations - Expander

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Upsampling -expanding in time -compressing in frequency

≠

Interchanging Operations - Expander

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Upsampling -expanding in time -compressing in frequency

=

Interchanging Operations - Compressor

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Downsampling -compressing in time -expanding in frequency

=

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=

Interchanging Operations - Compressor

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=

Interchanging Operations - Compressor

=

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=

Interchanging Operations - Compressor

=

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= After

compressing

Interchanging Operations - Compressor

=

Interchanging Operations - Summary

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Filter and expander Expander and expanded filter*

Compressor and filter Expanded filter* and compressor

*Expanded filter = expanded impulse response, compressed freq response

Multi-Rate Signal Processing

!  What if we want to resample by 1.01T? "  Expand by L=100 "  Filter π/101 ($$$$$) "  Downsample by M=101

!  Fortunately there are ways around it! "  Called multi-rate "  Uses compressors, expanders and filtering

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Polyphase Decomposition

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!  We can decompose an impulse response (of our filter) to:

Polyphase Decomposition

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!  We can decompose an impulse response (of our filter) to:

Polyphase Decomposition

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Polyphase Decomposition

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Polyphase Decomposition

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Polyphase Decomposition

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Polyphase Decomposition

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Polyphase Implementation of Decimation

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!  Problem: "  Compute all y[n] and then throw away -- wasted

computation! "  For FIR length N # N mults/unit time

Polyphase Implementation of Decimation

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Polyphase Implementation of Decimation

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Interchanging Operations - Summary

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Filter and expander Expander and expanded filter

Compressor and filter Expanded filter and compressor

Polyphase Implementation of Decimation

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Polyphase Implementation of Decimation

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Each filter computation: -N/M multiplications -1/M rate per sample #N/M*(1/M) mults/unit time

Total computation: -M filters #N/M mults/unit time

Multi-Rate Signal Processing

!  What if we want to resample by 1.01T? "  Expand by L=100 "  Filter π/101 ($$$$$) "  Downsample by M=101

!  Fortunately there are ways around it! "  Called multi-rate "  Uses compressors, expanders and filtering

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Polyphase Implementation of Decimator

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interpolator decimator

Polyphase Implementation of Interpolation

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interpolator decimator

E0(z)

E0(z)

E0(z)

Multi-Rate Filter Banks

!  Use filter banks to operate on a signal differently in different frequency bands "  To save computation, reduce the rate after filtering

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Multi-Rate Filter Banks

!  Use filter banks to operate on a signal differently in different frequency bands "  To save computation, reduce the rate after filtering

!  h0[n] is low-pass, h1[n] is high-pass "  Often h1[n]=ejπnh0[n] $ shift freq resp by π

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Multi-Rate Filter Banks

!  Assume h0, h1 are ideal low/high pass

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Multi-Rate Filter Banks

!  Assume h0, h1 are ideal low/high pass

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Multi-Rate Filter Banks

!  Assume h0, h1 are ideal low/high pass

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Multi-Rate Filter Banks

!  Assume h0, h1 are ideal low/high pass

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Have to be careful with

order!

Multi-Rate Filter Banks

!  Assume h0, h1 are ideal low/high pass

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Multi-Rate Filter Banks

!  Assume h0, h1 are ideal low/high pass

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Multi-Rate Filter Banks

!  h0, h1 are NOT ideal low/high pass

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Non Ideal Filters

!  h0, h1 are NOT ideal low/high pass

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Non Ideal Filters

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Perfect Reconstruction non-Ideal Filters

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Quadrature Mirror Filters

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Quadrature mirror filters

Big Ideas

!  Downsampling/Upsampling !  Practical Interpolation !  Non-integer Resampling !  Multi-Rate Processing

"  Interchanging Operations

!  Polyphase Decomposition !  Multi-Rate Filter Banks

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Admin

!  HW 4 due Friday "  Typo in code in MATLAB problem, corrected handout "  See Piazza for more information

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