When choosing an interpolation technique, pulse-shaping or multi-stage interpolation methods should be considered.

David A. Hall, National Instruments

Designers of today's software-defined radios and test instrumentation use a variety of digital signal processing (DSP) techniques to improve system performance. Interpolation is one DSP technique that can be used to increase the sample rate of digital signals and it is commonly implemented for several reasons. In transceivers

click to enlarge

Figure 1. Time and frequency domain of signal after zero Insertion.
using homodyne (direct) upconversion, interpolation can actually improve analog performance. Moreover, it is often required in heterodyne (using an intermediate frequency) upconversion to enable mixing baseband signals with a digital carrier. This is called digital upconversion. Thus, interpolation has several practical purposes in modern communications systems.

In this article, we will illustrate various interpolation techniques, address special considerations when choosing an interpolation method, and explain several common reasons why interpolation is performed. The specific interpolation methods covered include: linear, zero-insertion, zero-order-hold and frequency domain stuffing.

Linear Interpolation Method
The first technique, linear interpolation, is conceptually one of the simplest methods of increasing a signal's sample rate. Using this method, a linear fit is applied between each pair of existing samples. Then, new samples are derived according to the linear fit and inserted between each pair of original samples. To interpolate a signal by N times, N-1 new samples must be inserted between each pair of original samples. As one might imagine, the algorithm for linear interpolation is rather simplistic to implement. However, the computation required for applying a linear fit between each sample is expensive compared to other methods. Thus, while it is used occasionally, linear interpolation is not always the preferred interpolation method.

Zero-Insertion Method
A second method to interpolate a signal is the “zero-insertion” technique. With this technique, the sample rate of a waveform is increased by inserting zeros between each original sample. The process of zero insertion increases the sample rate of the original signal relative to its fundamental frequency content. In order to increase the sample rate of a waveform by the order of ‘N', a total of 'N — 1‘ zero samples must be inserted between each original sample. Zero-insertion of a signal is illustrated in Figure 1, which shows both the time and frequency domain of a signal after zero-insertion.

As we can see in the frequency domain, the process of zero insertion creates distortion products that are centered at multiples of the original sample rate. Thus, from our original sinusoid at 10 MHz, we will see new images at 90 MHz, 110 MHz, 190 MHz, etc. (Note that in this particular simulation, we have added a small amount of noise to the signal to simulate a more realistic real-world environment).

While zero-insertion introduces distortion at higher frequencies, it does not introduce distortion within the frequency band of interest. Thus, a low-pass filter (Figure 2) can be used to remove the

click to enlarge

Figure 2. A zero stuffed signal after low-pass filter is applied..
distortion products without producing distortion in the original signal bandwidth. Typically, a digital finite impulse response filter with a cutoff at 0.5 × Original Sample rate is used to remove distortion products. This is illustrated in Figure 2.

As the figure illustrates, the low-pass filter attenuates the distortion products and restores the shape of the original signal. The filter used in this simulation is a 200 tap FIR with a lower pass band of 48 MHz and a lower stop band of 50 MHz. In the time domain, we are able to observe an interpolated signal in which the shape of the original sinusoid has been restored. Note that because zero-insertion increases the power of the fundamental tone by a factor of 1/N, digital gain must be applied to restore the signal to its original amplitude.

Zero-Order-Hold Method
The zero-order-hold technique of interpolation works in much the same way as the previous method, except with one small caveat. While the zero-insertion method operates by adding zeros between each original data point, the zero-order-hold method simply repeats each sample. Similar to zero-insertion, N-1 copies of the original sample must be inserted to increase the sample rate by a factor of N.

The effect of repeating samples also introduces distortion at higher frequencies. With this method, the pre-filtered signal contains images centered at multiples of the original sample rate. When interpolating a sinusoid at 10 MHz, these images occur at 90-, 110-, 190 MHz, etc. Again, these images can simply be filtered using a digital FIR with a cutoff less than 0.5 × Original Sample Rate.

While the zero-order-hold method is computationally similar to the zero-insertion method, it does have one distinct disadvantage. Zero-order-hold interpolation does introduce smaller distortion products at multiples of the original sample rate. Thus, the filtering requirements are slightly less stringent. However, unlike the zero-order-hold method, it also introduces a small amount of distortion in the band of interest. Because these distortion products cannot be removed by a low-pass filter, the zero-insertion method is generally preferred in communications systems.

FFT Expansion Method
The final method for signal interpolation is known as FFT expansion. With this technique, interpolation is performed by affecting the frequency domain instead of the time domain. To perform FFT Expansion, a Fast Fourier Transform (FFT) of the signal is first computed to return a frequency domain representation of the

click to enlarge

Figure 3. FFT of signal with zeros inserted in frequency domain.
signal. Then, the frequency domain is expanded by adding zero-power samples to represent power at higher frequencies. In order to interpolate our signal by an order of N, (N-1) * FFT size / 2 samples must be prepended to the front of our FFT result, and (N-1) * FFT size / 2 samples must be appended to the end of the FFT result. This is illustrated in Figure 3, which shows how one can increase the increases the sample rate of the waveform relative to its frequency. With this method, we can return an interpolated signal simply by performing an inverse FFT of the expanded frequency domain.

As with the previous methods, FFT expansion does reduce the power of the fundamental signal relative to the order of interpolation. When interpolating by an order N, the power is reduced by a factor of 1/N. Thus, digital gain must also be applied to restore the signal to its original power. While the FFT Expansion method does not typically require a low-pass filter to restore the shape of the original signal, it is computationally quite expensive. As a result, it is not generally used in communications systems.

Performance Benefits from Multi-Stage Filters
One important consideration when choosing an interpolation technique is the consideration of pulse-shaping or multi-stage interpolation methods. In general, the overall processing load can be reduced by combining interpolation with pulse shaping or by performing interpolation in multiple sequential stages. Pulse shaping filters operate essentially as a low-pass filter by limiting the bandwidth of a signal. Thus, interpolation through the zero-insertion or zero-order-hold techniques can be implemented using a pulse-shaping finite impulse response filter (PFIR) at the low-pass filter stage of interpolation. This saves significant computation because filtering does not need to be applied twice. A second common practice when optimizing performance is to interpolate the signal in stages with sequential interpolation blocks. Because the cost computation increases significantly with the order of interpolation, interpolating the signal in stages can reduce overall processing time. As an example, three successive interpolation blocks of N = 4 is equivalent to one interpolation block of order N = 64. However, the second method is much more computationally expensive. In practical applications, it is also common to use multiple filtering techniques in the multi-stage approach. As an example, a pulse-shaping finite impulse response filter (PFIR) can be used in succession with a cascade integrator comb (CIC) filter. This technique provides an optimal tradeoff between accuracy and efficiency because computationally efficient techniques such as the CIC can be combined with low-distortion techniques such as FIR zero-insertion.

Improvements in System Performance
Increasing the sample rate of digital baseband waveforms is necessary for several reasons in software defined radios. First, it can improve the analog performance and reduces analog filtering requirements. Second, interpolation enables sample rate conversion for devices using multiple communication standards. Finally, interpolation is often required to enable digital upconversion.

Increasing the sample rate of baseband waveforms is important because today's digital-to-analog converters use a sample-and-hold method of producing analog signals. While the sample-and-hold technique is able to produce a close approximation representation of a truly analog signal, the sample-and-hold

click to enlarge

Figure 4. Block diagram of digital upconversion in a vector signal generator.
characteristics of a DAC also introduce spectral images at multiples of the update rate. Interpolation addresses this issue by pushing images farther away from the bandwidth of interest. This relaxes the requirements for analog filtering because a higher frequency cutoff with better passband flatness can be used. As an added benefit, interpolation improves the linearity of the signal being generated. Because sample-and-hold digital-to-analog converters follow a (Sin X)/X amplitude response, signals at the edge of the DAC bandwidth will be slightly distorted. By increasing the sample rate relative to the signal bandwidth, the affect of (Sin X)/X roll off is reduced.

Second, interpolation can be combined with successive decimation or interpolation stages to enable sample rate conversion. Today's wireless devices must support multiple communications standards, and they must also be capable of generating baseband signals at different symbol rates. Because baseband signals must be sampled at a rate that is an integer multiple of the symbol rate, sample rate conversion is necessary to use a constant update rate for the digital to analog converter.

Finally, increasing the sample rate of baseband waveforms enables digital upconversion of narrowband signals to an intermediate frequency. Digital upconversion is the process of mixing baseband I and Q signals to a digital carrier. As part of digital upconversion, baseband I and Q signals are mixed with a digital in-phase and quadrature-phase carrier. Moreover, it requires upsampling the baseband waveforms to a sample rate that can be used to represent the digital carrier.

As Figure 4 illustrates, baseband I and Q signals must pass through a pulse-shaping and interpolation stage before being mixed (mathematically represented as multiplication) with the numerically controlled oscillator (NCO), which is the digital IF carrier.

It is important to note that the use of DSP techniques such as digital upconversion or digital downconversion is not limited to wireless devices alone. In fact, test instrumentation such as National Instruments PXI RF vector signal generators and vector signal analyzers use these DSP techniques as a mechanism for data reduction. It is these very technologies that enable 20 MHz PXI RF record and playback applications.

We hope that you can appreciate some of the creative signal processing tricks that can be implemented to perform digital interpolation. However, it is important to not lose sight of the fact that interpolation is often an important aspect of digital communication system design. Whether it is used simply to remove baseband signal images or perform digital upconversion, this DSP technique is widely used both in wireless devices and in RF test instrumentation.

About the Author
David A. Hall is product marketing engineer for RF Instruments at National Instruments, (888) 280-7645;