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m28 - Upsampling, Signal Stretching and Interpolation

Module by: C. Sidney Burrus. E-mail the author

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Summary: Properties of the Fourier transform can be used to describ the effects of sampling, stretching, and interpolating.

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Up--Sampling, Signal Stretching, and Interpolation

In several situations we would like to increase the data rate of a signal or, to increase its length if it has finite length. This may be part of a multi rate system or part of an interpolation process. Consider the process of inserting M − 1 M − 1 zeros between each sample of a discrete time signal.

y ( n ) = { x ( n / M ) < n > M = 0  (or n = kM) 0 otherwise y ( n ) = { x ( n / M ) < n > M = 0  (or n = kM) 0 otherwise (1)
For the finite length sequence case we calculate the DFT of the stretched or up--sampled sequence by
C s ( k ) = ∑ n = 0 M N − 1 y ( n ) W M N n k C s ( k ) = ∑ n = 0 M N − 1 y ( n ) W M N n k (2)
C s ( k ) = ∑ n = 0 M N − 1 x ( n / M ) ⨿ M ( n ) W M N n k C s ( k ) = ∑ n = 0 M N − 1 x ( n / M ) ⨿ M ( n ) W M N n k (3)
where the length is now N M N M and k = 0 , 1 , ⋯ , N M − 1 k = 0 , 1 , ⋯ , N M − 1 . Changing the index variable n = M m n = M m gives:
C s ( k ) = ∑ m = 0 N − 1 x ( m ) W N m k = C ( k ) . C s ( k ) = ∑ m = 0 N − 1 x ( m ) W N m k = C ( k ) . (4)
which says the DFT of the stretched sequence is exactly the same as the DFT of the original sequence but over M M periods, each of length N N .

For up--sampling an infinitely long sequence, we calculate the DTFT of the modified sequence in ((Reference)) as C s ( ω ) = ∑ n = − ∞ ∞ x ( n / M ) ⨿ M ( n ) e − j ω n = ∑ m x ( m ) e − j ω M m C s ( ω ) = ∑ n = − ∞ ∞ x ( n / M ) ⨿ M ( n ) e − j ω n = ∑ m x ( m ) e − j ω M m

= C ( M ω ) = C ( M ω ) (5)
where C ( ω ) C ( ω ) is the DTFT of x ( n ) x ( n ) . Here again the transforms of the up--sampled signal is the same as the original signal except over M M periods. This shows up here as C s ( ω ) C s ( ω ) being a compressed version of M M periods of C ( ω ) C ( ω ) .

The z-transform of an up--sampled sequence is simply derived by:

Y ( z ) = ∑ n = − ∞ ∞ y ( n ) z − n = ∑ n x ( n / M ) ⨿ M ( n ) z − n = ∑ m x ( m ) z − M m Y ( z ) = ∑ n = − ∞ ∞ y ( n ) z − n = ∑ n x ( n / M ) ⨿ M ( n ) z − n = ∑ m x ( m ) z − M m (6)
= X ( z M ) = X ( z M ) (7)
which is consistent with a complex version of the DTFT in (Equation 5).

Notice that in all of these cases, there is no loss of information or invertibility. In other words, there is no aliasing.

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