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In the following document, learn how factors such as bit resolution and sampling rate affect the ability of a signal generator to produce an ideal analog signal.
This tutorial is part of the National Instruments Signal Generator Fundamentals series. Each tutorial in this series will teach basic concepts about the architecture, features, or applications of signal generators.
While the fundamental building block of analog signal generation is a Digital-to-Analog Converter (DAC), the process of generating advanced analog signals is much more complex. In this document, we will discuss the factors that affect signal quality including: bit resolution, bandwidth, sample rate, interpolation, and filtering
Digital to Analog Conversion Characteristics
National Instruments signal generators, like many in the industry, utilize off the shelf Digital-to-Analog Converters (DAC’s) from vendors such as Analog Devices. While many factors affect the quality of a waveform from a signal generator, three fundamental characteristics profoundly affect its performance when generating an analog waveform. These three characteristics are: bit resolution, bandwidth, and sampling rate.
Bit Resolution
The bit resolution of a signal generator is limited by the bit resolution of the DAC that it uses. This specification of a signal generator is important because it affects the amplitude range of the device. Fundamentally, a DAC is able to approximate the actual analog signal by outputting series of voltages levels at discrete analog levels. The bit resolution of an ADC is important because it determines the number of discrete voltage levels that are possible, and thus the voltage difference between each one.
An example, consider a 3-bit DAC. With three bits of resolution, this DAC divides its vertical range into 23 or 8 discrete analog voltage levels. These 8 discrete voltage levels are divided across the entire vertical range and the voltage difference between each level is similar. Thus, when a 3-bit DAC is configured with an output range of 0-10 V, each level accounts for (10 V / 8) = 1.25 volts. Thus, at this range, a 3-bit DAC cannot generate voltage differences smaller than 1.25 V.
On the other hand, consider a 16-bit DAC with the same output range. With 16 bits of resolution, this DAC is able to generate 216 or 65,536 discrete voltage levels. In addition, with the same 10 V vertical range, the signal generator is able to generate voltage differences as small as 153 µV. Thus, we are able to more accurately approximate a waveform simply by increasing the bit resolution of the DAC. The difference between a 3-bit DAC with a 16-bit DAC is illustrated in the figure below.
Figure 1: The Effect of Bit Resolution on Amplitude Accuracy
In this figure, the black line represents a waveform that has been generated with a 16-bit DAC. Zoomed out, this appears as an ideal analog sine wave. However, if we zoomed in we would be able to see that it is composed of small, discrete voltage steps. On the other hand, the blue line represents a sine wave that has been generated with a 3-bit DAC. As the graph illustrates, a 3-bit DAC is not able to approximate an analog signal with much precision. Although both waveforms are composed of a series of discrete voltage levels, the 16-bit DAC is able to approximate the analog signal much more precisely.
To find the smallest discrete step size of a waveform output from a signal generator, we can use the following formula.
With this formula, ‘Range’ is the maximum output range of the signal generator. This typically depends on the output path selected. Common maximum ranges can be as low as ±1V for the direct path of a signal generator to ±6V for a high gain signal path. Thus, when 16-bit signal generator is configured for a 2 V peak-to-peak output, the discrete step size is 30.6 µV.
Bandwidth
The bandwidth of a signal generator represents the maximum frequency that it can generate without significant attenuation. More specifically, bandwidth is defined as the maximum frequency at which the output is attenuated by 3 dB or less, relative to the amplitude of a DC or low frequency signal. Typically, bandwidth is limited by the specific DAC chosen and the design of the signal generator circuitry. In addition, it is an important specification because it determines both the maximum frequency of sine output, and specifications such as overshoot and rise time for the instrument.
To illustrate the importance of bandwidth for square wave generation, we contrast the overshoot and rise time of a 5 MHz square wave that has been generated with two different signal generators. As we can observe from the figure below, the square wave generated by the signal generated with 80 MHz bandwidth shows a smaller overshoot and faster rise time. Thus, bandwidth is an important specification to characterize.
Figure 2: Bandwidth comparison for 5 MHz Square Wave
One way that bandwidth can be characterized is with a bode plot. Below, we show a bode plot for the NI 5422 signal generator, which graphs the amplitude (dB) versus the frequency (MHz).
Figure 3: Bode Plot Illustrating PXI-5422 Bandwidth
As the figure illustrates, the output of the signal generator becomes attenuated as the signal increases in frequency. Note a line has been drawn -3dB from the desired amplitude of the test signal. The intersection of this line with the bode plot defines the bandwidth of the device. On this particular signal generator, the bandwidth is just over 80 MHz.
Sample Rate
The sample rate of a signal generator also determines the maximum frequency that can be generated. As we mentioned earlier, a DAC is able to approximate analog signals by generating discrete voltage values over time. Thus, the sample rate of a signal generator is the rate at which the DAC can be updated to generate a new voltage level. Ideally, a DAC should have the highest sample rate possible to precisely represent high-frequency signals.
The Nyquist criteria demand that for sinusoids, the sample rate must be at least twice the frequency of the signal to accurately represent its frequency. In addition, a sine wave must be sampled at 2.5 times its frequency to accurately represent its phase as well. Moreover, to represent the shape of a signal, it should be sampled at least 10 times its frequency. To illustrate this, we first show a sine wave that has been sampled a 2.5x its frequency:
Figure 4: Sampling at 2.5x Sinusoid Frequency
By sampling at 2.5x the frequency of the signal, it is possible to accurately represent the signals frequency and phase. However, in order to represent the shape of the signal, we will need to sample it much faster. Below, we show the same signal when it has been sampled at 10x its frequency:
Figure 5: Sampling at 10x Sinusoid Frequency
As the graph above illustrates, higher sample rates enable us to more closely approximate an analog signal with a DAC. Thus, to generate a high-frequency signals, it is important to use a high sample rate. Note that sinusoidal signals, it is more appropriate to consider the effective sample rate of a signal generator. Because these signals are smooth in nature, interpolation can greatly improve representation of the signal by increasing the sample rate.
Filtering and Interpolation
The Nyquist criteria merely provide guidelines for the sampling rate required to generate a given signal. However, it does not account for the other considerations required to generate a truly analog signal. Because digital-to-analog converters use a sample-and-hold technique, generating even a highly oversampled signal still produces high-frequency images. Below, we illustrate the time domain of a sinusoid that has been sampled at 20x its frequency. Note that the sample and hold characteristics of a 16-bit DAC are evident in “stepped” appearance of the waveform.
Figure 6: Sample-and-hold Output of a Signal Generator
While this approximation of the sinusoid works well visually, the actual output is not spectrally pure. We can observe this effect in the frequency domain.
Digital Filtering (Interpolation)
Because DAC’s use a sample-and-hold output mechanism, they typically create high frequency images of the original sinusoid. These images occur at each multiple of the sampling rate, plus or minus the fundamental tone. Thus, when generating a 20 MHz sinusoid sampled at 100 MHz, you will see images at 80 MHz, 120 MHz, 180 MHz, 220 MHz, etc. Below, we show the frequency domain of the 20 MHz sine wave.
Figure 7: Spectral Images of a 20 MHz Sine Wave
As the graph illustrates, high-frequency spectral images can distort the frequency domain of the signal being generated. Thus, when a single tone is required, these images must be removed.
National instruments signal generators utilize an analog and/or digital filter to remove high-frequency images. First, a digital finite-impulse response filter (FIR) is able to provide interpolate the signal to increase the effective sample rate. For example, suppose a 20 MHz sine wave is sampled at 100 MS/s and interpolated by 4x to an effective sample rate of 400 MS/s. By increasing the effective sample rate, the nearest spectral images of are moved to center around the new effective sampling rate as shown below:
Figure 8: Spectral Images of a 20 MHz Sine Wave with 4x Interpolation
As the figure above illustrates, digital filtering (interpolation) is not able to eliminate spectral images entirely. Instead, it merely shifts them to a higher frequency.
Analog Filtering
However, many signal generators also implement an analog filter as well. The analog filter is able to attenuate these spectral images below the noise flor. This is illustrated in the image below, which shows the same frequency domain once a low-pass analog filter has been applied:
Figure 9: 20 MHz Sine Wave with Interpolation and Analog Filter
As the figure above illustrates, the images around 400 MHz have been dropped to below the noise floor of the device. In this specific example the analog low-pass filter has been designed to attenuate high-frequency images by as much as 60 dB. As a result, the signal generator is able to produce an analog signal that is a more accurate approximation of an idea analog signal. Thus, we can observe the time-domain of the interpolated and filtered signal in the figure below:
Figure 10: Time Domain of 20 MHz Sine Wave
As the image above illustrates, the individual steps once evident in the time domain are no longer visible. Instead, the output appears like a pure sinusoid.
As this document has discussed, factors such as bit resolution, bandwidth, sampling rate, interpolation, and analog filtering all contribute to the ability of a signal generator to accurately approximate an analog signal. For a more thorough tutorial on how filtering and interpolation affects spectral purity, see the following document: Filtering and Interpolation to Improve Spectral Purity
Related Links
ni.com/signalgenerators
ni.com/modularinstruments
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