Frequency Response of Linear Interpolation

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Linear Interpolation as Resampling

Convolution Interpretation

Triangular Pulse as Convolution of Two Rectangular Pulses

To downsample a signal by the factor N means to form a new signal consisting of every Nth sample of the original signal. — Click for https://ccrma.stanford.edu/~jos/mdft/Downsampling_Operator.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

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The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Linear Interpolation as Resampling

Convolution Interpretation

Triangular Pulse as Convolution of Two Rectangular Pulses

Frequency Response of Linear Interpolation

Since linear interpolation can be expressed as a convolution of the samples with a triangular pulse, we can derive the frequency response of linear interpolation. Figure 4.7 indicates that the triangular pulse serves as an anti-aliasing lowpass filter for the subsequent downsampling by . Therefore, it should ideally ``cut off'' all frequencies higher than .

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Frequency Response of Linear Interpolation

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University