Multirate digital signal

processing

Prasanta Kumar Ghosh

Oct23, 2018

Reconstruct and resample

If has bandwidth less than , then if

Note the infinite summation in the reconstruction (not practical)

Suppose for sampling rate conversion, we need to sample at

with a desired sampling rate of

thus obtained are accurate only if

If , components above should be filtered out before

resampling. if

If , then it is a convolution summation (LTI system)

For

Let

Integer Fraction

Time-varying system for sampling rate conversion

**Inefficient when interpolating function is complicated

It simplifies when where D and I are relatively prime integers

Value of mD

modulo I

Thus, can take only I distinct values

Hence, only I distinct impulse responses are possible and thus, is

periodic

A linear and periodically time-varying

discrete-time system (great simplification!)

Downsampling/Decimation

Only one impulse response for all m

Upsampling/Interpolation

Impulse response is shifted

at an increment of

Downsampling/Decimation

Upsampling/Interpolation

Fractional shifting results in

Other way is to create a new sequence by inserting zero samples

Decimation by a factor D

Is decimation LTI operation?

Decimation by a factor D

Decimation by a factor D

Decimation by a factor D

Evaluate the Z-transform on unit circle with frequency variable

Thus, gets stretched to by down-sampling

If is correctly designed, then aliasing is eliminated and

Decimation by a factor D

Interpolation by a factor I

DTFT:

Interpolation by a factor I

As the frequency component of

are unique in the range

Images beyond that in

should be rejected by low pass

filtering

C = ?

Interpolation by a factor I

is the desired normalization factor

Sampling rate conversion by a rational factor I/D

Sampling rate conversion by a rational factor I/D

Sampling rate conversion by a rational factor I/D

Frequency response of the combined filter