The top trace in Figure 21 shows a pseudo-random noise record of 256K points and a 50 MHz clock. The key clock-related feature is that the waveform appears to be a staircase function. The sharp edges can be removed by the AWG's 20 MHz low-pass filter, as shown in the second trace above. The TDS' FFT spectra for the two signals are overlaid below the two time-domain waveforms. The salient characteristic of the unfiltered noise spectrum is that it rolls off with a (sin x)/x function with the first null at the clock frequency and subsequent nulls at multiples of the clock rate. Even below the first null, the noise density is constant only well below the clock frequency. In this example, we used the AWG's 20 MHz hardware filter to attenuate the frequency components above the first spectral null to achieve a bandwidth limited noise stimulus with relatively constant noise density from DC to a few MHz.

Figure 21. Upper waveform is a pseudo-random sequence with a 50 MHz clock. The second waveform is the same signal filtered by the AWG's 20 MHz low-pass filter. The lower two traces are the calculated FFT spectra. Note that the horizontal scale of the spectra is 20 MHz/div.

* In this paper, AWG always refers to Arbitrary Waveform Generator, as opposed to Additive White Gaussian, an unfortunate coincidence in this context.

Verifying Read Channel Frequency Response

A popular application of a noise source is to quickly verify the frequency response through various points in the read channel chain. The top trace in Figure 22 shows a 20 MHz bandpass filter's output response to an AWG noise waveform. The FFT of the filter output is shown in the second trace. If the AWG noise waveform had a constant noise density, then we could simply use the FFT of the output signal as our indicator of the channel's response. But the AWG noise waveform has its characteristic downward slope starting at DC (third trace).

Figure 22. The top trace is a bandpass filter's response to an AWG noise waveform. The AWG noise signal was clocked at 1 GS/s and low-pass filtered at 100 MHz. The second trace is the FFT of the output signal. The third trace shows the downward sloping FFT magnitude of the clocked noise waveform. The TDS calculates the actual filter response by subtracting the FFT of the filter output from the FFT of the noise stimulus.

Thus, to correct for the 4 dB drop in noise density across the DC-to-50-MHz frequency span of interest, we must use the TDS' math function to subtract the filter's output spectrum from the noise input spectrum (bottom trace).

Figure 23 compares the calculated filter response with the response as measured by a conventional swept network analyzer. While the swept analyzer offers superior resolution and higher dynamic range, it is a costly, specialized tool that is not always available. The AWG/TDS combination is an effective alternative for quickly verifying channel filter performance.

Figure 23. A comparison of TDS calculated filter response (from Figure 22) using the AWG's noise function against the response using a conventional swept network analyzer (smooth line).

Simulating Signal-to-Noise Ratios

The AWG's noise waveform can be added to our read channel waveforms to simulate different signal-to-noise ratios. The first step is to create a waveform record using the AWG's noise function. In Figure 24, a 960-point noise record is generated, which we will add to the 960-point 1F pattern shown in Figure 15.

Figure 24. The noise waveform is specified in terms of its peak amplitude.

Figure 24a. The histogram of the 960-point noise waveform sorted into the 256 bins. Although a random number generator produces uniformly distributed values, the AWG processes the output of its random number generator so that its characteristics more closely match a Gaussian noise source.

As mentioned previously, the AWG simulates a noise waveform by calculating a sequence of random numbers. Digital random number generators typically produce uniformly distributed values, but circuit noise is better modeled with a Gaussian distribution. In practice, the AWG actually calculates a noise value by averaging 12 consecutive random numbers. Thus, by the central-limit theorem, the noise values will more closely approximate a Gaussian distribution than the underlying uniform distribution. Figure 24a shows a histogram of the 960-point noise waveform. There are 256 bins ranging from ­127 to +128, each corresponding to a level of the AWG's 8-bit D/A converter. The histogram increasingly approaches a Gaussian distribution as the record length increases. Unlike a true normal distribution, note that the tails of the AWG noise distribution abruptly cut off at the specified peak value. This peak value is entered as the amplitude of the noise waveform. The crest factor, or peak to RMS ratio, is approximately 3 to 1.

Figure 25 shows the resulting waveform as displayed by the AWG's waveform editor.

Figure 25. The 960-point noise waveform is added to the 960-point 1F waveform using the AWG's dual waveform math function. The result is a 960-point waveform. Marker pulses were added to provide a convenient trigger pulse once per cycle.

Making Coherent Noise More "Random"

Figure 26. The top trace is the 960-point noise waveform (19.2 mV rms). The middle trace is the 1F waveform with noise (270 mV rms). The bottom trace is the FFT of the waveform.

Figure 26 shows the 960-point noise waveform and the 1F waveform plus noise as captured by the TDS. The wideband signal-to-noise ratio is 20*log(270Ž19) or 23 dB. The bottom trace is the FFT of the waveform which gives us more revealing information. While the ~10 MHz 1F fundamental and its odd harmonics are the dominant components, there are clearly discrete spectral components rather than the desired flat noise floor. The problem lies in the selection of the record length for the noise waveform. As discussed in the beginning of this section, a 960-point noise waveform will repeat every 960 points. In this case, the AWG clock rate was 1 GS/s, making the noise pattern repeat every 960 ns or at about 1 MHz. Furthermore, the noise pattern is pre-calculated with the 1F pattern such that the two waveforms are forever coherent. This is an unrealistic test signal to simulate a noisy channel.

With a pre-calculated signal plus noise waveform, we cannot completely escape the fact that the two components are implicitly correlated. We do not want to change the fundamental 1F frequency of ~10 MHz since this is the signal component. We can, however, reduce the effect of the correlation between the signal and noise components by decreasing the repetition rate of the noise waveform. For example, we can simply multiply the record lengths by 250 so we have 240,000-point waveforms instead of 960-point waveforms. The new 1F record is the same fundamental signal repeated over a longer interval. But the 240,000-point noise waveform repeats 250 times less often than the 960-point noise waveform clocked at the same rate.

Figure 27. When the waveform record length is increased to 240,000 points, the noise density better approximates a level noise floor. The signal-to-noise ratio is actually the same (23 dB), but the longer record length provides a more realistic test signal.

Figure 27 shows that the discrete spectral noise components have been transformed into a more recognizable noise floor, and we now have a more suitable test signal. The longer noise record is still a repetitive waveform, but its noise density characteristics are better distributed in the frequency domain. The TDS' FFT measurement in Figure 26 and Figure 27 were derived from a time domain record sampled at 1 GS/s and a record length of 5000 points. Thus, each FFT calculation is based on a 5 µs snapshot of the test signal, but the noise pattern repeats only every 250 µs. This means that we need to average the FFT calculations over time to accumulate its characteristics. The bin size, or spacing between FFT frequency domain samples, is 200 kHz, or the ratio of the sample rate to the record length. The equivalent noise bandwidth of the FFT filter was about 2 bins. Thus, when averaged over time, the noise density at this frequency resolution is more evenly distributed with the longer record.