multi rate

Single-rate systems:

Sampling rates at the input and at the output and all internal nodes are the same

Multirate systems:

DSP systems with unequal sampling rates at various parts of the system

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Digital Audio : Three different samplingrates are employed

• Broadcasting : 32 KHz• Digital Compact Disc : 44.1 KHz• Digital Audio Tape : 48 KHz

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

Systems that employ multiple sampling rates in the processing of digital signals are called multirate digital signal processing systems.


Analog signal is sampled every TS secs. Ts is referred to as the sampling interval. fs = 1/Ts is called the sampling rate or

sampling frequency.


when the sampling rate is lower than the Nyquist rate, the reconstruction is not possible as the solution is not unique.

Aliasing frequencies

maxkff s

Where k is a positive or negative integer

maxkff s


Down sampling : by a factor of M is achieved by discarding M-1 samples for every M samples. x[

n]Mx(n) y(m)


Mfs fs/M

else.0

,if)()(

mMnnxmy


Mx(n) y(m)H(z)

fs fsM=fs/M

fc : fs/(2M)

Decimation is the process of decreasing the sampling rate by a factor of M i.e. from Fs to Fs / M


Upsampling by a factor L→ inserting L-1 zeros between two samples


Lfs Lfs

else.0

,if)()(

mLnmxny


Interploation is the Process of increasing the sampling rate of the signal by a factor of L i.e. from F s to LFs

Lx(m) y(n)H(z)

Lfsfs

fC : fs/2


Transferring Data from C.D to audio tape 44.1K → 48K

Transferring Data from Recording Studio to C.D 192K → 44.1K


Process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth

Advantages: Anti-aliasing filter has less complexity High resolution

Over Sampling


It is the process of digitization and direct down conversion of band limited signal located apart from that to base band location

It takes the advantages of aliasing

It reduces the speed requirements of ADC and hence reduces the amount of digital memory necessary

Band Pass Sampling


Noble Identities


Cascade Equivalences

• To implement a fractional change in the sampling rate we need to employ a cascade of an up-sampler and a down-sampler

• Consider the two cascade connections shown below

M L][nx ][1 ny

ML][nx ][2 ny


Cascade Equivalences

• A cascade of a factor-of-M down-sampler and a factor-of-L up-sampler is interchangeable with no change in the input-output relation:

• if and only if M and L are relatively prime, i.e., M and L do not have any common factor that is an integer k > 1


Polyphase Decomposition

• Polyphase decomposition is the decomposition of a sequence x(n) into sub-sequences x(nMN+i)

[ 0,1,2,3,4,5,6,7,8,9,10,11]

[ 0,3,6,9][ 1,4,7,10]

[ 2,5,8,11]

)(0 nx

)(1 nx

)(2 nx

M=3


Polyphase Decomposition


Filters in Multirate Systems

• The importance of filtering arises from the fact that the sampling theorem should be respected for all the sampling rates of the system at hand.

• Filters are required to bandlimit the spectrum of the signal to the prescribed bandwidth in accordance with the actual sampling rate.


FIR Structures


IIR Structures


FIR Decimators


FIR Interpolators


Polyphase Structures for FIR Decimator


Polyphase Structures for FIR Interpolator


IIR Decimator


IIR Interpolator


Two Channel Filter Banks


Quadrature Mirror Filter Banks

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