baudline solution - swept sine vs. WGN

Abstract

Swept sine vs. white Gaussian noise
E. Olson, May 6 2005

Develop a method for measuring the frequency response of an unknown propagation channel. Compare the benefits and drawbacks of using a linear sine wave sweep versus white Gaussian noise (WGN) as the stimulus signal. The goal is to create a fast and accurate procedure for determining the filter response of a full duplex black box DUT.

Introduction

Use the baudline signal analyzer to generate the sweep and WGN sources, capture the input signal, and calculate the average spectra. The following describes the theory and procedure needed to characterize the frequency response of a propagation channel.

black box
Treat the Device Under Test (DUT) as an unknown black box. The only way that information about the propagation channel can be extracted is by probing with an input stimulus and then measuring the output characterization signal. Standard black box testing consists of these three elements:

test signal generator (source)
black box (DUT)
signal analysis tool (sink)

The propagation channel can add various forms of noise and distortion to the input signal and this change in signal characteristics defines the black box filter function. The primary focus of this application note is how to determine the frequency response of the black box filter function.

test signal source

Baudline's Tone Generator is a versatile signal source that can create the appropriate stimulus functions needed for this study. The output functions are highly configurable and a digital loopback mode is even possible for quick experimental testing. Both white Gaussian noise and a swept linear sine wave will be investigated.

white Gaussian noise
WGN is constant spectral energy at all frequencies with a probability histogram that follows a Gaussian bell shaped curve (see above right). It is truly random noise with a flat frequency spectrum and that makes WGN an excellent excitation test signal.

Due to WGN's random nature, the calculated spectrum slices will have a large variance. If high resolution plots are desired then longer Average collection times will be necessary.

swept sine
The linear sine sweeps that are of interest here will start at 0 Hz (DC) and end at the Nyquist frequency. The full spectrum width is swept and this creates a constant spectral energy at all frequencies (just like WGN). The duration parameter is important. Too short of a duration creates a sweeping lobe that is too wide for any fine detail in the frequency domain. The selection of the spectrogram ON and OFF times are also important for absolute energy measurements. These issues make using the swept sine stimulus signal slightly more complicated than WGN. So careful experimental technique is required.

The tone generator settings for a 48000 sample rate:

start frequency: 0 Hz
end frequency: 48000 Hz
function: sine
modulation: sweep
shape: linear
direction: up
duration: 30 seconds
channel: left & right
gain: 0 dB

black box setup

The black box can be an external piece of sophisticated hardware or it can be a simple cable. It can even be a virtual concept such as the digital loopback mode. The setup may be different and vary greatly but the testing methods remain the same.

loopback wiring
We want to measure the performance of our test equipment. The quality of the ADC and the DAC play a very important role in test and measurement. They act as hard limits to the upper bounds of performance. So our first goal is to measure these limits and then use them to create a baseline standard to work from. An added bonus is doing this with a minimal amount of equipment by using a loopback mode. We are going to make the black box test itself by making the black box the cable. Or as is depicted in the picture above, the black box now becomes baudline!

With baudline there are four possible loopback modes:

external cable
internal "volume" mixer channel
digital tone generator loopback option
half duplex operation for special cases

The first three are proper full duplex modes where the sound card's input is connected to it's output. For half duplex an external function generator is required (this could be a second sound card running baudline). The half duplex mode is important for three reasons:

some older sound cards only operate at half duplex
sometimes the full duplex performance differs from the half duplex performance.
the same card can be run at two different sample rates for "Nyquist folding."

gain settings
Set the gains so that the test signal is strong enough to be far above the noise floor. The test signal should also be a bit below the clipping point so that the distortion measurements are good for all of the sweep frequencies. A high output gain and a low input gain are good starting points for finding the optimal gain settings. Having optimal gain is not as critical for this test experiment as it is for other tests but a strong signal with reduce the effects of the noise floor. So a strong signal that avoids excessive distortion should be the goal. Using the THD and SINAD distortion measurement windows are useful tools in finding a good working gain value.

Measuring frequency response at multiple gain levels can be insightful for discovering many different types of non-linear filter behavior. Also using a completely attenuated test signal (zero) is a good way of measuring the frequency response of the noise floor.

signal analysis

This section compares the frequency response qualities of a swept sine wave versus white Gaussian noise (WGN). Baudline's Tone Generator is used as the signal source and the spectrum are collected with the Average window. For a fair comparison both the swept sine duration time and the WGN collection time are set for 30 seconds.

The audio card for the following test is the Creative Sound Blaster 128 (also known as the SB128 and PCI16 ENS1371).

swept sine and WGN
For this test the input and the output sample rates were both set to 48000 and a single instance of baudline was running in full duplex mode.

The WGN test source was collected for 30 seconds in the Average window as the red plot. WGN is a random signal and the jaggedness is a function of the variance which is fairly large. A longer collection time would result in a smoother plot.

Pressing the Manual Trigger button started a 30 second duration linear sweep from 0 Hz to 24000 Hz (the Nyquist frequency) after which baudline was paused. The start and end sections of the sweep were carefully selected in the spectrogram and then pasted into the Average window.

Legend:

red - WGN
green - sine sweep

Other than the 11 dB offset, the swept sine and the WGN curves look very similar. To correct for the dB offset the above swept sine test was rerun with a -11 dB digital gain setting in Tone Generator window. The dB axis was zoomed in 32X to get a better look. Below is the result.

The above zoomed in plot makes the 3 dB of passband filter ripple easy to measure an it makes it easy to compare the two different test functions. The swept sine and WGN curves match perfectly except for variance errors due to WGN's short collection time. WGN needs to be collected for about 8 minutes (a 16X increase) before it starts to approach the smoothness of the swept sine. Note that variance is proportional to the square root of time.

The gain difference between the swept sine and WGN brings up another very important point. Frequency resolution gain is a function of sample rate, decimation, and/or FFT size. This phenomena is described with more rigor in the Sine Distortion Measurement application note. What this means is that as the bin resolution increases the difference between the swept sine and WGN gain also increases. This can cause problems because WGN gets pushed down towards the noise floor and the closer the signal is to the noise floor the larger the noise floor's impact can be.

calibration
The system frequency response can be calibrated with baudline's Equalization window. This equalization produces a flat spectrum and it removes any frequency response errors due to the ADC. Both WGN and the swept sine average spectral responses can be used as a correction source by pressing the Grab Average button. The Auto Collect button can also be used for calibration in the Record and the Pause modes.

Nyquist folding
The goal of this section is to build a complete picture of the audio card's anti-alias low pass filter (LPF). The previous experiment measured only the in-band Nyquist response and the objective here is to measure the out-of-band response. Usually this information is only available to the hardware designer but with a little clever use of aliasing the stop-band filter response can be extracted.

The Nyquist sampling theorem determines the highest frequency that can be represented by a particular sampling rate. Signals that are beyond the Nyquist frequency limit either get attenuated by the ADC or DAC's LPF and/or they get aliased. This aliasing is called the "fold back."

See the spectrogram image on the right. Two instances of baudline were run; the Tone Generator source at a 48000 sample rate, and the collection sink at a 24000 sample rate. A zero Hz to 24000 Hz sine wave sweep was the source test signal.

The two lines of the swept "V" shape were selected individually and then pasted into the Average window. The 0 Hz to Nyquist samples make up the green spectrum. The folded back Nyquist to 0 Hz samples make up the purple spectrum. Below is the Average window which shows this:

Legend:

green - sine sweep
purple - sine sweep folded

The purple curve in the above plot was flipped around the Nyquist point to show the full LPF response. This was accomplished with a short program that operated on the saved Average window's ASCII text file. The filter stopband attenuation for the SB128 measures to be about -34 dB which not that great for a computer sound card.

Conclusion

Both the swept sine and WGN test signals generated good frequency response measurements. Now let's compare the benefits of each test method.

swept sine:

faster
less variance
better resolution
stronger so noise floor has less of an effect
can do Nyquist folding or out-of-band response
trickier to do correctly, thus more prone to error

WGN:

simple to do, just turn it on and accumulate
doesn't excite resonant modes like a sine wave
wideband spectral source less likely to cause non-linear effects

The best signal source to use depends a lot on your specific requirements. Actually the best answer is to use both. Baudline makes it easy to test with both swept sine and WGN sources. Comparing the differences between the two methods could yield useful insights that otherwise might of remained unknown.