Upsampling

[1][2][3] 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.

Rate increase by an integer factor

can be explained as a 2-step process, with an equivalent implementation that is more efficient:[4] 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:[a]

sequence is the impulse response of the interpolation filter, and

The interpolation filter output sequence is defined by a convolution: The only terms for which

can be designed as a half-band filter, where almost half of the coefficients are zero and need not be included in the dot products.

phases of the impulse response is filtering the same sequential values of the

In some multi-processor architectures, these dot products are performed simultaneously, in which case it is called a polyphase filter.

times faster than the original input rate.

zeros between the useful outputs of a phase and adding them to a sum is effectively decimation.

That equivalence is known as the second Noble identity.

Then the discrete-time Fourier transform (DTFT) of the

sequence is the Fourier series representation of a periodic summation of

{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x[n]}\ e^{-i2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X{\Bigl (}f-{\frac {k}{T}}{\Bigr )}.}

An example of both these distributions is depicted in the first and third graphs of Fig 2.

[6] When the additional samples are inserted zeros, they decrease the sample-interval to

Omitting the zero-valued terms of the Fourier series, it can be written as: which is equivalent to Eq.2, regardless of the value of

That equivalence is depicted in the second graph of Fig.2.

, which increases the number of periodic spectral images within the new bandwidth.

[7]  The second graph also depicts a lowpass filter and

resulting in the desired spectral distribution (third graph).

The filter's bandwidth is the Nyquist frequency of the original

but filter design applications usually require normalized units.

(see Fig 2, table) Let L/M denote the upsampling factor, where L > 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.

cycles per intermediate sample, is the lower frequency.

Fig 1: Depiction of one dot product, resulting in one output sample (in green), for the case L=4, n=9, j=3. Three conceptual "inserted zeros" are depicted between each pair of input samples. Omitting them from the calculation is what distinguishes a multirate filter from a monorate filter.
Fig 2: The first triangle of the first graph represents the Fourier transform X ( f ) of a continuous function x(t) . The entirety of the first graph depicts the discrete-time Fourier transform of a sequence x[n] formed by sampling the continuous function x(t) at a low-rate of 1/T . The second graph depicts the application of a lowpass filter at a higher data-rate, implemented by inserting zero-valued samples between the original ones. And the third graph is the DTFT of the filter output. The bottom table expresses the maximum filter bandwidth in various frequency units used by filter design tools.