Real-time spectrum analysis basics

FFT returns N equally spaced frequency domain samples of X[k]. The amplitude of X[k] is shown in Figure 2-14. Note that the samples returned by FFT may miss the amplitude peak of X[k].

CZT can return M frequency domain samples with arbitrary start and end frequencies (Figure 2-15). Note that CZT does not change the underlying frequency domain output of the DFT. It simply extracts a different set of frequency domain samples than the FFT.

The advantage of using a CZT is that the frequencies of the first and last samples in the frequency domain can be chosen arbitrarily, independent of the input sampling rate. The same results can also be achieved by arbitrarily controlling the input sampling rate so that the output of the FFT results in the same output samples as the CZT. In both cases, the end result is the same. The choice is purely a matter of implementation, and both options may be suboptimal solutions depending on the requirements and available hardware.

Figure 2-16 above. Frequency response of the low-pass filter. Figure 2-17 below. Pulse response of the low-pass filter in Figure 2-15.

Digital filtering

Many applications use frequency filters to select certain frequencies and reject others. Traditional filters are implemented using analog circuit cells (RLC), while DSP mathematically selects the frequencies to be enhanced or attenuated. A common mathematical implementation is the Finite Impulse Response (FIR) filter. RSAs use FIR filters throughout. In addition to common signal conditioning applications that require specific frequency bands to be passed or rejected, FIR filters can also be used to adjust for non-ideal characteristics of analog signal paths. Internally generated alignment data is combined with stored factory calibration data to create a FIR filter whose response compensates for the frequency response of the analog signal path, allowing the cascade of the analog and digital paths to have a flat amplitude response and linear phase.

Frequency Response and Impulse Response

The Fourier Transform theorem states that there is equivalence between the frequency domain and the time domain. It further tells us that the transfer function of a device (usually expressed as a magnitude and phase response that varies with frequency) is equal to the impulse response during the measurement. FIR filters use a discrete time approximation with a finite time period to simulate the impulse response of the desired filter transfer function. Signal filtering is then performed by convolving the input signal with the filter's impulse response.

Figure 2-18. Filter multiplication by frequency response. Figure 2-19. Convolution in the time domain is equivalent to multiplication in the frequency domain.

Numerical convolution

The frequency domain is often used to analyze the response of linear systems, such as filters. The signal is represented by its frequency components. The spectrum of the signal at the filter output is calculated by multiplying the input signal spectrum by the filter's frequency response. Figure 2-18 illustrates frequency domain operations. Fourier's theorem states that multiplication in the frequency domain is equivalent to convolution in the time domain. The frequency domain multiplication in the figure above is equivalent to convolving the time domain representation of the input signal with the filter's impulse response, as shown in Figure 2-19.

Figure 2-20. Discrete-time numerical convolution.

All frequency filters require memory cells. Reactive cells commonly used in analog filters have memory because their output in the circuit depends on the current input and the input at various points in time. A discrete-time filter can be constructed using actual memory cells as shown in Figure 2-20.

The lower register is used to store the impulse response of the filter, with earlier samples on the right and later samples on the left. The upper register is used to shift the input signal from left to right, once per clock cycle. The contents of each corresponding register are multiplied together, and all the products are summed once per clock cycle, and the summed result is the filtered signal.

Figure 2-21 a, b. Comparison: (a) Swept spectrum analyzer MaxHold curve after 120 seconds; (b) Tektronix real-time spectrum analyzer, DPX bitmap MaxHold curve after 20 seconds.

In summary, RSA relies heavily on digital signal processing for spectrum analysis. Key points about DSP for RSA include: RSA6000 uses a combination of FFT and CZT to achieve spectrum display.

• The FFT is more computationally efficient and can achieve faster transform rates, while the CZT is more flexible and can provide variable resolution bandwidth for a fixed set of input samples.

• The resolution bandwidth (RBW) shape is achieved by applying an optimized window function to the time domain signal before performing a Fourier transform. The RBW is specified by the 3 dB bandwidth and the 60 dB:3 dB shape factor in the same way as in the analog implementation. In general, the shape factor of the digitally implemented filter is lower (dramatically) than that of the analog implementation, making it easier to resolve closely spaced signals with very different amplitudes.

By applying an optimized window function, other form factors can be used for specific applications.

The RSA3000 Series RSAs use a combination of methods when performing spectrum analysis.

• In Spectrum mode, the result of the windowed FFT is convolved with the RBW shape to produce a Spectrum curve with the specified RBW, similar to an analog spectrum analyzer. This process results in a shape factor of approximately 5:1, slightly wider than the 4.1:1 of the RSA6000.

• In DPX mode, CZT is used to resolve bandwidth flexibility.

• In RSA mode, using a windowed FFT, the noise bandwidth is specified in the typical way for FFT analysis. The noise bandwidth is approximately 6% (0.25 dB) higher than the RBW.

In this section we have seen that digitally implemented correction and filtering are key factors in implementing the high conversion rates required by the RSA. In the next section we will examine how these filters can be effectively used in the unique display provided by the RSA - the digital phosphor spectrum display.

DPX Technology: A Revolutionary Signal Discovery Tool

Tektronix patented Digital Phosphor Technology or DPX reveals signal details that are completely missed by traditional spectrum analyzers and VSAs (Figure 2-20). DPX's real-time RF display of the spectrum shows you signals you haven't seen before, allowing users to see signals instantly, greatly speeding up the discovery and diagnosis of problems. DPX is a standard feature on all Tektronix RSAs.

Digital fluorescent display

The term "digital phosphor" comes from the phosphor that coats the cathode ray tube (CRT) used in televisions, computer monitors, and other test equipment displays. When the electron beam activates the phosphor, it fluoresces, illuminating the path drawn by the electron stream.

Although raster scan technologies such as liquid crystal displays (LCDs) eventually replaced CRTs in many applications due to their thickness and low power advantages, the combination of phosphor layers and vector displays in CRTs offers several advantages for modern test and measurement applications.

Afterglow: Afterglow is the continued emission of light even after the electron beam has passed. Normally, fluorescence fades quickly enough that it is not noticeable to the viewer, but even very small amounts of afterglow can still allow the human eye to detect events that are too brief to be seen.

Proportionality: The slower the electron beam passes through a point on the phosphor screen, the brighter the resulting light. As the frequency of the electron beam increases, the brightness of a point also increases. The user can intuitively understand how to interpret this Z-axis information: bright parts of the track indicate frequent events or slow electron beam movement; dark tracks come from infrequent events or fast-moving electron beams.

Instruments that use LCDs (or raster CRTs) and digital signal paths do not inherently offer persistence and proportionality. Tektronix developed digital phosphor technology to achieve the analog advantages of vector CRTs, and digital oscilloscopes and now RSAs have further improved on this technology. Digital enhancements such as brightness grading, selectable color schemes, and statistical traces convey more information in less time.

Figure 2-22. Example of a 3D bitmap database after 1 update (left) and 9 updates (right). Note that each column contains the same total number of "triggers".

DPX Display Engine

The simplest description of what DPX technology does in an RSA is that it performs thousands of spectrum measurements per second, updating the screen at real-time rates. It takes thousands of acquisitions per second and transforms them into a spectrum. This high transform rate is critical for detecting infrequent events, but it is too fast for an LCD to keep up with, and too fast for the human eye to perceive. So the incoming spectrum is written at full speed into a bitmap database, which is then transferred to the screen at a rate that the human eye can see. This bitmap database can be thought of as a dense grid of a spectrum plot divided into rows representing the amplitude values ​​of the trace and columns representing points on the frequency axis. Each cell in this grid contains the number of triggers that entered the spectrum. By tracking these counts, digital phosphor technology achieves proportionality, allowing the user to visually distinguish rarely occurring transients from common signals and background noise.

An actual 3-D database in RSA contains hundreds of columns and rows, but we will use an 11X10 matrix to illustrate the concept. The left figure in Figure 2-21 illustrates what a database cell might contain after plotting a spectrum. A blank cell has a value of zero, meaning that no point in the spectrum falls within it.