Professor A G Constantinides 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] INPUTOUTPUT

Professor A G Constantinides 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

Professor A G Constantinides 3 Interpolation & Decimation Or Where So that

Professor A G Constantinides 4 Interpolation & Decimation Hence the structure may be realised as + OUTPUT INPUT Samples across here are phased by T secs. i.e. they do not interact in the adder. Can be replaced by a commutator switch.

Professor A G Constantinides 5 Interpolation & Decimation Hence INPUT Commutator OUTPUT

Professor A G Constantinides 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

Professor A G Constantinides 7 Interpolation & Decimation Set And Where in both expressions the subsequences are constructed as earlier. Then

Professor A G Constantinides 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

Professor A G Constantinides 9 Interpolation & Decimation The structure realising this is + INPUT OUTPUT Commutator

Professor A G Constantinides 10 Interpolation & Decimation For FIR filters why Downsample and then Upsample? #MULT/ACC LENGTH N LOW PASS LENGTH N LOW PASS LENGTH N LOW PASS DOWNSAMPLE M:1UPSAMPLE 1:M #MULT/ACC TOTAL #MULT/ACC

Professor A G Constantinides 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.

Professor A G Constantinides 12 Interpolation & Decimation For the impulse response shown The amplitude response is then given In general

Professor A G Constantinides 13 Interpolation & Decimation Now consider Then

Professor A G Constantinides 14 Interpolation & Decimation Hence Also Or For a normalised response

Professor A G Constantinides 15 Interpolation & Decimation Thus The shifted response is useful

Professor A G Constantinides 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

Professor A G Constantinides 17 Multistage Design of Decimator and Interpolator Specifications for the decimation filter H(z) are assumed to be as follows:,,,

Professor A G Constantinides 18 Polyphase Decomposition The Decomposition Consider an arbitrary sequence {x[n]} with a z-transform X(z) given by We can rewrite X(z) as where

Professor A G Constantinides 19 Polyphase Decomposition The subsequences are called the polyphase components of the parent sequence {x[n]} The functions, given by the z-transforms of, are called the polyphase components of X(z)

Professor A G Constantinides 20 Polyphase Decomposition The relation between the subsequences and the original sequence {x[n]} are given by In matrix form we can write

Professor A G Constantinides 21 Polyphase Decomposition A multirate structural interpretation of the polyphase decomposition is given below

Professor A G Constantinides 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:

Professor A G Constantinides 23 Polyphase Decomposition Its 4-branch polyphase decomposition is given by where

Professor A G Constantinides 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

Professor A G Constantinides 25 Polyphase Decomposition Example - Consider To obtain a 2-band polyphase decomposition we rewrite H(z) as Therefore, where

Professor A G Constantinides 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

Professor A G Constantinides 27 Polyphase Decomposition Example - Consider the transfer function of a 5-th order Butterworth lowpass filter with a 3-dB cutoff frequency at 0.5  : It is easy to show that H(z) can be expressed as

Professor A G Constantinides 28 Polyphase Decomposition Therefore H(z) can be expressed as where

Professor A G Constantinides 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)

Professor A G Constantinides 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):

Professor A G Constantinides 31 FIR Filter Structures Based on Polyphase Decomposition A direct realization of H(z) based on the Type I polyphase decomposition is shown below

Professor A G Constantinides 32 FIR Filter Structures Based on Polyphase Decomposition The transpose of the Type I polyphase FIR filter structure is indicated below