Digital Signal Processing

Soma Biswas

2017

𝑃𝑎𝑟𝑡𝑖𝑎𝑙 𝑐𝑟𝑒𝑑𝑖𝑡 𝑓𝑜𝑟 𝑠𝑙𝑖𝑑𝑒𝑠: 𝐷𝑟.𝑀𝑎𝑛𝑜𝑗𝑖𝑡 𝑃𝑟𝑎𝑚𝑎𝑛𝑖𝑘

Example

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Example

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Linear convolution of two finite-length sequences

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Therefore, (𝐿 + 𝑃 − 1) is the

maximum length of the sequence

𝑥3[𝑛] resulting from the linear

convolution of a sequence of length 𝐿with a sequence of length 𝑃.

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Time aliasing in the circular

convolution of two finite length

sequences can be avoided if 𝑁 ≥ 𝐿 +𝑃 − 1, Also it is clear that if 𝑁 = 𝐿 =𝑃, all of the sequence values of the

circular convolution may be different

from those of the linear convolution.

Contd.

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Contd.

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Contd.

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Implementing LTI system using DFT

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• Since LTI systems can be implemented by convolution, circular convolution can be used to implement these systems.

• Lets us first consider an L-point input sequence 𝒙[𝒏] and a P-point impulse response 𝒉[𝒏]. The linear convolution of these two sequences, which will be denoted by 𝒚[𝒏], has finite duration length (𝑳 + 𝑷 − 𝟏).

• If a circular convolution is done with at least (𝑳 + 𝑷 − 𝟏) points, it will be identical to linear convolution.

• The circular convolution can be achieved by multiplying the DFTs of 𝒙[𝒏] and 𝒉[𝒏]. Both 𝒙[𝒏] and 𝒉[𝒏] must be augmented with sequence values of zero amplitude. This process is often referred to as zero-padding.

• The output of a FIR system whose input also has finite length can be computed with DFT.

• In many applications, such as filtering of speech waveform, the input signal is of indefinite duration. While, theoretically, we might be able to store the entire waveform and then implement the procedure using DFT for a large number of points, however, such DFT is generally impractical to compute.

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• Another drawback is that in this method, no filtered output samples can be computed until all the input samples have been collected. Generally we would like to avoid such delay in processing.

• The solution to both problems is block convolution, in which the signal to be filtered is segmented into sections of length 𝑳.

• Each section can then be convolved with the finite-length impulse response and the filtered sections fitted together in an appropriate way.

• The linear filtering of each block can then be implemented using the DFT.

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We assume 𝑥 𝑛 = 0 𝑓𝑜𝑟 𝑛 < 0 and

the length of 𝑥[𝑛] is much greater

than 𝑃.

The sequence 𝑥[𝑛] can be

represented as a sum of shifted finite-

length segments of length 𝐿; i.e.,

Convolution Is LTI operation

DCT

Soma Biswas

DSP 2017

DFT

DCT

• Basis sequences are cosines – are periodic and have even symmetry

• -> Extension of x(n) outside (0,N-1) in synthesis equation will be periodic & symmetric

• DFT: finite length sequence -> form periodic sequences

• DCT: finite length sequence -> form periodic, symmetric sequences –many ways of doing this

• Most popular is DCT-2

DCT 2

• DCT-2 Transform pair

Relation between DCT-2 and DFT

Relation between DCT-2 and DFT

Provides fast algorithms can be used to

compute DFT and so DCT

Relation between DCT-2 and DFT

Energy Compaction Property

• Used in many data compression - property of “Energy Compaction”

• DCT-2 of a finite length sequence has its coefficients more highly concentrated as low indices than DFT

• From Parseval’s theorem

• Many of the later DCT coeff. Can be set to 0 without significant impact on the energy of the signal

Energy Compaction Property