In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. Upsampling can be synonymous with expansion, or it can describe an entire process of expansion and filtering (interpolation). When upsampling is performed on a sequence of samples of a signal or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate (or density, as in the case of a photograph). For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample rate is 55,125.

Upsampling by an integer factor

can be explained as a 2-step process, with an equivalent implementation that is more efficient:

1. Expansion: zeros. A notation for this operation is:

2. Interpolation: Smooth out the discontinuities using a lowpass filter, which replaces the zeros.

In this application, the filter is called an interpolation filter, and its design is discussed below. When the interpolation filter is an FIR type, its efficiency can be improved, because the zeros contribute nothing to its dot product calculations. It is an easy matter to omit them from both the data stream and the calculations. The calculation performed by a multirate interpolating FIR filter for each output sample is a dot product:

sequence is the impulse response is non-zero. The interpolation filter output sequence is defined by a convolution:

Thus: and the convolution can be rewritten as:

can be designed as a half-band filter such subsequences (called phases sequential output values. In some multi-processor architectures, these dot products are performed simultaneously, in which case it is called a polyphase filter.

zeros between the useful outputs of a phase and adding them to a sum is effectively decimation. It's the same result as not computing them at all. That equivalence is known as the second Noble identity. It is sometimes used in derivations of the polyphase method.

Interpolation filter design

be the Fourier transform sequence. Then the discrete-time Fourier transform sequence is the Fourier series representation of a periodic summation

has units of hertz (Hz)

which is also the desired result of interpolation. An example of both these distributions is depicted in the first and third graphs of Fig 2.

Omitting the zero-valued terms of the Fourier series, it can be written as:

which is equivalent to , which increases the number of periodic spectral images within the new bandwidth. Some authors describe that as new frequency components. resulting in the desired spectral distribution (third graph). The filter's bandwidth is the Nyquist frequency sequence. but filter design applications usually require normalized units. (see Fig 2, table)

Upsampling by a fractional factor

Let L/M denote the upsampling factor, where L>M.

1. Upsample by a factor of L

2. Downsample by a factor of M

Upsampling requires a lowpass filter after increasing the data rate, and downsampling requires a lowpass filter before decimation. Therefore, both operations can be accomplished by a single filter with the lower of the two cutoff frequencies. For the L>M cycles per intermediate sample, is the lower frequency.

See also

Notes

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References

Further reading

• (discusses a technique for bandlimited interpolation)