Interpolation

1
Interpolation Decimation

• Sampling period T , at the
  output
• Interpolation by m
• Let the OUTPUT be i.e. Samples exist at
  all instants nT
• then INPUT is i.e. Samples exist
  at instants mT

2
Interpolation Decimation

• Let Digital Filter transfer function be then
• Hence is of the form i.e.
  its impulse response exists at the instants mT.
• Write

3
Interpolation Decimation

• Or
• Where
• So that

4
Interpolation Decimation

• Hence the structure may be realised as

INPUT
OUTPUT
Samples across here are phased by T secs. i.e.
they do not interact in the adder.
Can be replaced by a commutator switch.
5
Interpolation Decimation

• Hence

6
Interpolation Decimation

• Decimation by m
• Let Input be (i.e. Samples exist at
  all instants nT)
• Let Output be (i.e. Samples exist at
  instants mT)
• With digital filter transfer function
  we have

7
Interpolation Decimation

• Set
• And
• Where in both expressions the subsequences are
  constructed as earlier. Then

8
Interpolation Decimation

• Any products that have powers of less than
  m do not contribute to , as this is
  required to be a function of .
• Therefore we retain the products

9
Interpolation Decimation

• The structure realising this is

Commutator
10
Interpolation Decimation

• For FIR filters why Downsample and then Upsample?

11
Interpolation Decimation

• A very useful FIR transfer function special case
  is for N odd, symmetric
• with additional constraints on to
  be zero at the points shown in the figure.

12
Interpolation Decimation

• For the impulse response shown
• The amplitude response is then given
• In general

13
Interpolation Decimation

• Now consider
• Then

14
Interpolation Decimation

• Hence
• Also
• Or
• For a normalised response

15
Interpolation Decimation

• Thus
• The shifted response
• is useful

16
Design of Decimator and Interpolator

• Example Develop the specs suitable for the
  design of a decimator to reduce the sampling rate
  of a signal from 12 kHz to 400 Hz
• The desired down-sampling factor is therefore M
  30 as shown below

17
Multistage Design of Decimator and Interpolator

• Specifications for the decimation filter H(z) are
  assumed to be as follows
• , ,
  ,

18
Polyphase Decomposition

• The Decomposition
• Consider an arbitrary sequence xn with a
  z-transform X(z) given by
• We can rewrite X(z) as
• where

19
Polyphase Decomposition

• The subsequences are called the
  polyphase components of the parent sequence
  xn
• The functions , given by the
  z-transforms of , are called the
  polyphase components of X(z)

20
Polyphase Decomposition

• The relation between the subsequences and
  the original sequence xn are given by
• In matrix form we can write

21
Polyphase Decomposition

• A multirate structural interpretation of the
  polyphase decomposition is given below

22
Polyphase Decomposition

• The polyphase decomposition of an FIR transfer
  function can be carried out by inspection
• For example, consider a length-9 FIR transfer
  function

23
Polyphase Decomposition

• Its 4-branch polyphase decomposition is given by
• where

24
Polyphase Decomposition

• The polyphase decomposition of an IIR transfer
  function H(z) P(z)/D(z) is not that straight
  forward
• One way to arrive at an M-branch polyphase
  decomposition of H(z) is to express it in the
  form by multiplying P(z)
  and D(z) with an appropriately chosen polynomial
  and then apply an M-branch polyphase
  decomposition to

25
Polyphase Decomposition

• Example - Consider
• To obtain a 2-band polyphase decomposition we
  rewrite H(z) as
• Therefore,
• where

26
Polyphase Decomposition

• The above approach increases the overall order
  and complexity of H(z)
• However, when used in certain multirate
  structures, the approach may result in a more
  computationally efficient structure
• An alternative more attractive approach is
  discussed in the following example

27
Polyphase Decomposition

• Example - Consider the transfer function of a
  5-th order Butterworth lowpass filter with a 3-dB
  cutoff frequency at 0.5p
• It is easy to show that H(z) can be expressed as

28
Polyphase Decomposition

• Therefore H(z) can be expressed as
• where

29
Polyphase Decomposition

• In the above polyphase decomposition, branch
  transfer functions are stable allpass
  functions (proposed by Constantinides)
• Moreover, the decomposition has not increased the
  order of the overall transfer function H(z)

30
FIR Filter Structures Based on Polyphase
Decomposition

• We shall demonstrate later that a parallel
  realization of an FIR transfer function H(z)
  based on the polyphase decomposition can often
  result in computationally efficient multirate
  structures
• Consider the M-branch Type I polyphase
  decomposition of H(z)

31
FIR Filter Structures Based on Polyphase
Decomposition

• A direct realization of H(z) based on the Type I
  polyphase decomposition is shown below

32
FIR Filter Structures Based on Polyphase
Decomposition

• The transpose of the Type I polyphase FIR filter
  structure is indicated below