Multi Rate Digital Signal Processing

The process of employing multiple sampling rates in the processing of digital signals is called ‘Multi Rate Digital Signal Processing’.

The process of converting a signal from a given rate or sampling frequency to a different rate or sampling frequency is called ‘sampling rate conversion’.

Sampling Rate Conversion MethodsThere are two general methods are there for accomplishing sampling rate conversion of a digital signal, one method is,

Pass the digital signal through DAC, filter it if necessary, and then resample the resulting analog signal at desired rate i.e pass the analog signal through an ADC.

Disadvantages with this methodSignal distortion is introduced

i. by the DAC during signal reconstruction

ii. By the quantization effects during ADC

Another method is digital domain method.

In digital domain the sampling rate conversion can be done,

By means of ‘Down sampling’ - Reducing the sampling rate by an integer factor ‘D’.

By means of ’ Up sampling’ - Increasing the sampling rate by an integer factor ‘I’.

The process of reducing the sampling rate by an integer factor ‘D’ is called ‘Decimation’.

The process of increasing the sampling rate by an integer factor ‘I’ is called ‘Interpolation’.

Decimation by a factor D• Let consider x(n) be a i/p sequence which is passed

through a LPF, characterized by the impulse response denoted by h(n) and a frequency denoted by HD(ω) for performing decimation process as shown below:

LPFh(n)

Down sampler↓D

x(n) v(n)

• The o/p of the filter is a sequence given as,

Which is then down sampled by the factor D to produce y(m). Thus

• The frequency domain characteristics of the o/p sequence y(m) can be obtained by

• Now the Z-Transform of y(m) is

If the filter is properly designed then

[ ]c n[ ]c n[ ]c n[ ]c n

Standard ApproachDecimation by a Factor D

Interpolation by a factor I• The process of increasing the sampling rate of a digital

signal by an integer factor I is called Interpolation.

• In Interpolation process an I-1 new samples which are zero’s are interpolated between the successive values of the digital signal.

• Let v(m) be a sequence of sampling rate Fy=Ifx is obtained from x(n) after adding I-1 zero’s between successive values of x(n) and is expressed as,

↑I hI(m)v(m) y(m)

HI(ωy)

• The z-transform of v(m) is,

• The corresponding frequency spectra is obtained by substituting

↑I hI(m)v(m) y(m)

HI(ωy)

Sampling rate conversion by a rational factor I/D

• Sampling rate conversion by a rational factor ‘I/D’ can be achieved by first performing interpolation by the factor ‘I’ and then decimating the interpolator o/p by a factor ‘D’.

• In this process both the interpolator and decimator are cascaded as shown in the figure below:

Rate=IFx

Upsampler↑I

LPFhU(l)

x(n)Rate Fx

Interpolator

Downsampler

↓DLPFhd(l)

y(m)

Rate=Fx(I/D)=Fy

Decimator

• In this figure the two filters with impulse responses hu(l),hd(l)are operated at the same frequency=IFx and hence these two filters are combined in to a single LPF of impulse response h(l) which is shown in the figure below:

Rate=IFx

Upsampler↑I

LPFh(l)

x(n)Rate Fx

Downsampler

↓Dy(m)

Rate=Fx(I/D)=Fy

v(l) w(l)

H(ωv)

• Let v(l) be the o/p of the interpolator and can be represented as,

• Let w(l) be the o/p of the filter and can be obtined as,

• Finally the o/p of the sampling rate converter denoted by y(m) can be obtained as,

• The corresponding frequency domain representation is,=

FIR Filter • In general, a FIR system is described by the difference

equation

• or by the system transfer function

• According to the equ…(1)

y(n)=h(0)x(n)+h(1)x(n-1)+…….+h(M-1)x(n-M+1) and can be realized as

X(n) y(n)

y(n)=h(0)x(n)+h(1)x(n-1)+…….+h(M-1)x(n-M+1) and can be realized as

Filter Design and Implementation for Sampling Rate Conversion

• Here sampling rate conversion which is ‘Decimation’ and ‘Interpolation’ is performed by direct form FIR filter structures.

• The design and implementation of FIR filter for performing decimation process as shown in the figure:

• This realization is simple but inefficient because,

1.up sampling process introduces

I-1 zero’s between successive

points of the signal.

2.If ‘I’ is large, most of the signal components in the FIR filter are zero.

3.The multiplications and additions in the FIR filter result in zero’s due to this large ‘I’.

• Therefore it is necessary to develop a more efficient structure.

• This can be achieved by embedding the down sampling operation within the filter it self as shown in the figure.

• In this structure all multiplications and additions are performed at the lower sampling rate Fx/D.

• Thus desired efficiency can be achieved.

• Next consider the interpolation process which can be performed by means direct form FIR filter structures as shown in the figure.

• This structure is realized by first inserting I-1 zero’s between the samples of x(n) and then filtering the sequence.

• The major problem in this realization is that the filter computations are performed at high sampling rate Ifx.

• This problem is solved by using transposed form of FIR filter and embedding the up sampler within the filter as shown in the figure.

• So all multiplications are performed at the lower rate Fx.

Design and Implementation of Poly Phase Filter Structures for Sampling Rate Conversion

• The sampling rate conversion which is ‘interpolation’ (‘decimation’) is also performed by means of poly phase filter structures as shown in the figure below which results in better computational efficiency than FIR systems.

• Here each sub filter is defined with unit impulse responses

pk(n)=h(k+nI) where

k=0,1,……I-1,

n=0,1,……K-1

• This structure is achieved by reducing the large FIR filter of length ‘M’ in to a set of smaller filters of length K=M/I where ‘M’ is selected to be a multiple of ‘I’.

• Here all sub filters are basically al pass filters of different phase characteristics and are arranged in a parallel form.

• The o/p of each filter can be selected by a commutator.

• The rotation of the commutator is in the counter clockwise direction.

• This filter structure performs computations at low sampling rate Fx.

• Next is the decimator which can be realized by transposing the Interpolator structure as shown below:

• Here each sub filter is defined with unit impulse responses

pk(n)=h(k+nD) where k=0,1,……D-1, n=0,1,……K-1

• This structure is achieved by reducing the large FIR filter of length ‘M’ in to a set of smaller filters of length K=M/D is an integer and ‘M’ is selected to be a multiple of ‘D’.

Applications of Multi-rate Signal Processing

• Design of Phase Shifters

• Interfacing of Digital Systems with different sampling rates

• Implementation of narrow band LPF’s

• Implementation of Digital filter banks

• Sub band coding of speech signals

• Quadrature mirror filters

• Transmultiplexers

• Oversampling A/D and D/A conversion

Design of Phase Shifters• Here a network is designed that delays the signal x(n) by a

rational fraction of a sampling interval Tx i.e d=(K/I)Tx, where ‘d’ is the delay.

• In the frequency domain this delay corresponds to a linear phase shift of the form Θ(ω)=-(K/I) ω.

• Let consider the system which performs both ‘interpolation’ and ‘decimation’ as shown below:

• The interpolator increases the sampling rate by a factor ‘I’.

• The LPF eliminates the images or frequency duplications in the spectrum of the interpolated signal.

• Next the o/p of the filter is delayed by ‘k’ samples at the sampling rate IFx.

• Then the delayed signal is decimated by factor D=I.

• Thus the desired delay of (K/I)Tx will be achieved.

Interfacing of Digital Systems with different sampling rates

• Let consider the interfacing of two digital systems ‘A’ and ‘B’ through a digital sample and hold block as shown below:

• The sapling rate of system ‘A’ is Fx and the sampling rate of system ‘B’ is Fy .

• The o/p of system is fed to an interpolator which increases the sampling rate by a factor ‘I’.

• The o/p of the interpolator which is sampling rate IFx is fed to a digital sample-and-hold system.

• The signals from the digital sample-and-hold system are read out in to system ‘B’ at a rate IFy of system ‘B’.

• Thus the desired interfacing is achieved and the o/p rate of digital sample-and-hold system is not synchronized with the i/p rate.

Implementation of Digital filter banks• Digital filter banks are categorized as two types based on

‘decimation’ and ‘interpolation’

1. Analysis filter banks

2. Synthesis filter banks

• An analysis filter bank consists of a set of filters with system function {Hk(z)} are arranged in parallel with i/p x(n) as shown below:

• The frequency response characteristics of this filter bank splits the signal in to a corresponding number of sub bands.

• Next, a synthesis filter bank consists of a set of filters with system function {Gk(z)} are arranged in parallel with i/p yk(n) as shown below:

• The o/p this filter bank are summed to form a synthesized signal x(n).

• These filter banks are often used for performing spectrum analysis and signal synthesis.

• When a filter bank is used for computing DFT of a sequence {x(n)} then the filter bank is called DFT filter bank.

• The analysis filter bank for computing DFT consists of ‘N’ filters of system function {Hk(z)} where {k=0,1,2,…….N-1}.

• If {Hk(z)} where {k=1,2,…….N-1} then the analysis filter bank is called uniform DFT filter bank and the corresponding frequency domain representation is

• The frequency response characteristics of the filters {Hk(z), k=1,2,…….N-1} are obtained by uniformly shifting the frequency response of the filter having system function {H0(z)} by multiples of ‘2π/N’ .

• In the time domain the impulse responses are expressed as,

• The uniform DFT analysis filter bank can be realized as shown below:

• In this structure the frequency components in the sequence {x(n)} are translated in frequency to low pass by multiplying x(n) with

and the resultant signals are passed through LPF’s with impulse responses denoted by h0(n).

• The resulting decimated signal can be expressed as

• Where Xk(m) are samples of DFT at frequencies ωk=2πk/N .

• The corresponding synthesis filter for each element in the filter bank can be viewed as shown below:

• The i/p signal sequences [Yk(m),k=0,1,……N-1] are up-sampled by a factor of I=D, filtered to remove the images or image frequency components and translated in frequency by, multiplication by the complex exponentials {exp(j2πnk/N), k=0,1,……N-1}.

• The resulting frequency translated signals from the N filters are then summed for obtaining v(n) as shown below:

• Where the factor 1/N is a normalizing factor. {yn(m)}represent samples of the inverse DFT sequence corresponding to {yk(m)}.

{g0(n)} is the impulse response of the interpolation filter.

Sub band coding of speech signals

Quadrature mirror filters

Two channel QMF Bank