Editing Upsampling (section)

==Interpolation filter design==
[[File:Spectral views of zero-fill and interpolation by lowpass filtering.pdf|thumb|400px|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.]]

Let <math>X(f)</math> be the [[continuous Fourier transform|Fourier transform]] of any function, <math>x(t),</math> whose samples at some interval, <math>T,</math> equal the <math>x[n]</math> sequence. Then the [[discrete-time Fourier transform]] (DTFT) of the <math>x[n]</math> sequence is the [[Fourier series]] representation of a [[periodic summation]] of <math>X(f):</math>{{efn-la
|[[#f.harris|Harris 2004]]. "2.2". p 23. fig 2.12 (top).
}}
{{Equation box 1|title=
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<math>\underbrace{ \sum_{n=-\infty}^\infty \overbrace{x(nT)}^{x[n]}\ e^{-i 2\pi f nT}}_{\text{DTFT}} = \frac{1}{T}\sum_{k=-\infty}^{\infty} X\Bigl(f - \frac{k}{T}\Bigr).</math> &nbsp; &nbsp;
|{{EquationRef|Eq.2}} }} }}

When <math>T</math> has units of seconds, <math>f</math> has units of [[hertz|hertz (Hz)]]. Sampling <math>L</math> times faster (at interval <math>T/L</math>) increases the periodicity by a factor of <math>L:</math>{{efn-la
|[[#f.harris|Harris 2004]]. "2.2". p 23. fig 2.12 (bottom).
}}
{{Equation box 1|title=
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<math>\frac{L}{T}\sum_{k=-\infty}^\infty X\left(f-k\cdot \frac{L}{T}\right),</math> &nbsp; &nbsp;
|{{EquationRef|Eq.3}} }} }}

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.<ref name=LiTan/>&nbsp; 

When the additional samples are inserted zeros, they decrease the sample-interval to <math>T/L.</math>  Omitting the zero-valued terms of the Fourier series, it can be written as:

:<math>\sum_{n=0, \pm L, \pm 2L,..., \pm \infty}{} x(nT/L)\ e^{-i 2\pi f nT/L}
\quad \stackrel{m\ \triangleq\ n/L}{\longrightarrow} 
\sum_{m=0, \pm 1, \pm 2,..., \pm \infty}{} x(mT)\ e^{-i 2\pi f mT},</math>

which is equivalent to {{EquationNote|Eq.2,}} regardless of the value of <math>L.</math>  That equivalence is depicted in the second graph of Fig.2.  The only difference is that the available digital bandwidth is expanded to <math>L/T</math>, which increases the number of periodic spectral images within the new bandwidth.  Some authors describe that as new frequency components.<ref name=Lyons/>&nbsp; The second graph also depicts a lowpass filter and <math>L=3,</math> resulting in the desired spectral distribution (third graph).  The filter's bandwidth is the [[Nyquist frequency]] of the original <math>x[n]</math> sequence.{{efn-ua|Realizable low-pass filters have a [[transition band]] where the response diminishes from near unity to near zero. So in practice the cutoff frequency is placed far enough below the theoretical cutoff that the filter's transition band is contained below the theoretical cutoff.
}}&nbsp; In units of Hz that value is <math>\tfrac{0.5}{T},</math>&nbsp; but filter design applications usually require [[Normalized frequency (unit)|normalized units]]. (see Fig 2, table)