Interpolation

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3.2.2 Interpolationv

Upsampling is the process to increase the number of points per unit time used to describe

a signal. The spectral content of the signal does not change; what does change is the spectral

separation between images of the original spectrum. When upsampling is employed,

no new information is added to the signal. The process of upsampling decreases the time

between samples of a signal. This process can be used for matching sampling rates between

two systems or as the last step before the DAC to help relax the requirements for the

reconstruction filter. If the signal shown in Figure 3.6, X(ω_x), were sampled at twice the

rate shown in the figure, its spectrum would appear as shown in Figure 3.7. Several methods

of interpolation are presented in this section: zero-insertion, zero-order-hold (ZOH),

zero-insertion and raised-cosine filtering, and fast Fourier transform (FFT) expansion.

Zero-Insertion Interpolation

In this method, zeros are inserted between samples of a signal, generating a new signal.

This new signal is then lowpass filtered, yielding an upsampled version of the original (see

Figure 3.8).

For the purpose of analysis, assume that the original signal is x(n) and that the goal is

to upsample it by a factor of I. I − 1 zeros are inserted between each pair of consecutive

samples of x(n), yielding v(m), which can then be defined as

Taking the Z-transform of v(m) and defining m´ = mI yields V (z):

From Equations 3.15 and 3.14,

Figure 3.8: Direct Implementation of an Interpolator.

To evaluate the DTFT of V (z), evaluate Equation 3.16 with z = e^jωy, yielding

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Using Equation 3.18, the range for ω_y becomes  Equation 3.18 implies

that the upsampled signal can be recovered by filtering the zero-inserted signal with an LPF

h _I  ( m ) whose cutoff frequency is less than 

For illustrative purposes, a MATLAB example of this process is considered. The upsampling

factor selected for this example is four. Figure 3.9 shows the original signal and

its zero-insertion version. Figure 3.10 shows the time signals’ respective spectrum. As

expected, the zero-insertion version contains an additional I −1 = 3 copies of the original

spectrum, or a copy every  radians. This signal was filtered with a 200-order LPF with

a cutoff frequency ofradians, resulting in an interpolated version of the signal seen in

Figure 3.11. Note on the normalized scale, the peak has moved closer to zero radians per

second for the upsampled signal compared to that of Figure 3.10a. The true frequency is

still the same but is lower when normalized with respect to the sample rate.

Figure 3.9: (a) Sinusoid and (b) Zero-Insertion Equivalent (I = 4).

Figure 3.10: (a) Spectrum of Sinusoid and (b) Zero-Insertion Equivalent (I = 4; note both

scales are with respect to input sample rate).

Figure 3.11: (a) Time and (b) Frequency Domain Representation of the Interpolated Signal

(I = 4; note both scales are with respect to the output sample rate).

Zero-Order-Hold Interpolation

In this method, space between two samples is filled with a sample generated using some

function, in this case a ZOH or zero-order interpolation. This method for upsampling is

similar to the zero-insertion method described in Section 3.2.2, but this method is computationally

more expensive and generates less distortion at the higher frequencies for a

given interpolation filter. In zero-order interpolation, a signal is upsampled by extending a

sample’s value over several samples, as seen in Figure 3.12, before lowpass filtering.

Figure 3.12 presents an example of ZOH upsampling at four times the original sampling

rate. Sample n from the original sequence x[n] is placed in samples v[4n], v[4n + 1],

v[4n + 2], and v[4n + 3] of the new sequence v[m] of size 4n. This addition of extra

points creates a significant amount of high-frequency distortion in the form of replicated,

attenuated images in the spectrum.

The zero-insertion signal, v(m), is generated using Equation 3.14. In the ZOH version

of this signal, can be generated by Equations 3.19 and 3.20. A ZOH can be considered

as the sum of the sampled signal plus I − 1 delayed copies of v(m), shown in Equation

3.20.

 

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Figure 3.12: Sample and Hold. (Note: Black dots are the sampled signal and gray dots

represent the held value.)

V (z) can be calculated as

Substituting e^jωy for z in Equation 3.22 yields Equation 3.24, or a lowpass filtered

version of the spectrum seen in Equation 3.17. The attenuated images resulting from this

process can be filtered out using an LPF with a cutoff frequency less than , where I is the upsampling

ratio.

Figure 3.13 shows a sinusoid and its zero-order upsampled version before filtering. The

high-frequency components generated by the ZOH operation are seen in Figure 3.14. A

significant amount of distortion is apparently generated by this procedure. However, given

a well oversampled signal, this distortion is well beyond the spectrum of interest, so it can

be filtered out using an LPF. Figure 3.15 shows the waveform resulting from the filtering

operation and its spectrum. Note that the spurious signal content has been considerably

reduced by the holding process compared to the zero-insertion interpolated signal for the

same filter order. The gain of the LPF can be adjusted to ensure equal power for the input

and output signals.

Note that the use of a ZOH for interpolation introduces distortion on the interpolated

signal beyond that introduced through the zero-insertion method. Compared to the zeroinsertion

spectrum, the ZOH spectrum has an extra sinc term that helps to attenuate the

high frequency distortion although it slightly distorts the desired portion of the interpolated

signal spectrum. This in-band distortion can be cancelled by using the proper passband

characteristics in the interpolation filter design.

Figure 3.13: (a) Sinusoid and (b) Zero-Order-Hold Equivalent.

Figure 3.14: (a) Frequency Domain Representation (Magnitude) of the Original Sinusoid,

(b) Zero-Order-Hold Interpolated Signal (scaled to output sample rate).

Figure 3.15: (a) Time and (b) Frequency Domain Representation (Magnitude) of Filtered

ZOH Interpolated Signal (scaled to output sample rate).

Zero-Insertion and Raised-Cosine

To minimize inter-symbol interference (ISI), the use of raised-cosine pulse-shaping is important.

If the upsampling of a pulse code modulation (PCM) signal is to be performed at

the transmitter, the upsampling and raised-cosine filtering can be combined to simplify the

overall design.

This combination is performed by using the zero-insertion interpolation method described

earlier but with a raised-cosine filter instead of a lowpass interpolation filter. In

practice, the LPF can be ignored since the raised-cosine filter has a smaller bandwidth than

the interpolating LPF, as indicated in Figure 3.16. One could view the process as a pulse

train that excites a filter that performs both pulse-shaping and interpolation.

Figure 3.17 shows an example of this application. A three-bit PCM sequence is sampled.

Each of these samples ismapped onto a one-dimensional constellation and upsampled

before modulation. Though not shown in Figure 3.17, the pulse-shaped waveform is now

ready to be modulated and transmitted.

Figure 3.16: Combined Upsampling and Raised-Cosine Filtering.

Fast Fourier Transform Expansion

The FFT lends itself well for upsampling a signal. The basic idea behind the FFT approach

is to construct what the FFT of an oversampled signal should be by using the FFT of the

original signal. Each sample at the output of an FFT is separated by  where F_s is the

sampling frequency and N is the number of points used in each block. Upsampling using

an FFT is performed by taking N frequency samples, zero-padding N(I − 1) consecutive

zeros between the positive and negative images, and performing an inverse FFT of the

augmented signal. The principle behind this process is to create the Fourier transform of

the desired signal by inserting zeros in the frequency domain representation of the signal.

The resulting signal contains NI samples but spans the same time interval as the original

signal.

Figure 3.18 shows the original signal and its upsampled version; the signal used in this

example is a Rayleigh fading envelope. Figure 3.19 shows the magnitude and phase of the

original spectrum and the padded magnitude and phase of the upsampled signal. Although

Figure 3.17: (a) Upsampled Time Domain Signal, (b) Interpolated Signal with Raised-

Cosine Pulse-Shaping.

 

this example is not directly applicable to the implementation of a software radio, it is useful

for the simulation of fading channels, where fades could span over several samples [33].

Assuming that a signal x(n) was periodic with a period of N samples, then the discrete

Fourier transform (DFT) of that signal is defined as

The inverse DFT (IDFT) of this signal would be

Figure 3.18: Rayleigh Fading Envelope.

Figure 3.19: Rayleigh Fading Envelope, Original and Upsampled Spectrum (Magnitude

and Phase).

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The result seen in Equation 3.26 is a description of the same spectral content as seen

in Equation 3.25, but the signal in Equation 3.26 is cyclical over NI samples; it is an

oversampled version of the original signal x(n).

Relevant NI products

Customers interested in this topic were also interested in the following NI products:

• RF and Communication Hardware and Software
  
• Other Modular Instruments (digital multimeters, digitizers, switching, etc...)
• LabVIEW Graphical Programming Environment

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