Real-time spectrum analysis basics

Real-time spectrum analysis

For spectrum analysis to fall into the real-time category, it must process all the information contained within the bandwidth of interest without gaps and uncertainty. The RSA must obtain all the information contained in the time domain waveform and convert it into a frequency domain signal. To accomplish this in real time, several important signal processing requirements must be met:

• Provides sufficient capture bandwidth to support analysis of signals of interest

• A sufficiently high ADC clock rate to capture bandwidth exceeding the Nyquist criterion

• A sufficiently long analysis interval to support the narrowest resolution bandwidth (RBW) of interest A sufficiently fast DFT transform rate that the RBW of interest exceeds the Nyquist criterion

• The DFT rate exceeds the Nyquist RBW criterion, which requires overlapping DFT frames

• The degree of overlap depends on the window function

• The window function depends on the RBW

The current RSA meets the real-time requirements for the Frequency Mask Trigger (FMT) listed above for frequency spans up to the maximum real-time acquisition bandwidth. Therefore, triggering on a frequency domain event takes into account all information contained in the selected acquisition bandwidth.

Discover and capture transient events

Another application of fast repetitive Fourier transforms is to discover, capture, and observe infrequent events in the frequency domain. A useful metric is the minimum event period that captures a single non-repeating event with 100% probability. The smallest event is defined as the narrowest rectangular pulse that can be captured with 100% certainty at a specified accuracy. Narrower events can be detected, but accuracy and probability may be reduced. Discovering, capturing, and analyzing transient events requires:

• Provides sufficient capture bandwidth to support analysis of signals of interest

• A sufficiently high ADC clock rate to capture bandwidth exceeding the Nyquist criterion

• An analysis interval long enough to support the narrowest resolution bandwidth (RBW) of interest

• A fast enough DFT transform rate that the RBW of interest exceeds the Nyquist criterion

The RSA6000 Series DPX spectrum mode can measure 292,000 spectra per second, and can detect RF pulses as short as 10.3 ms with full accuracy and 100% probability. The swept spectrum analyzer (SA) sweeps 50 times per second and requires pulses longer than 20 ms to detect pulses with 100% probability and full accuracy.

Figure 2-7. Spectra, spectrogram, and sweep.

RSA vs. Swept Spectrum Analyzer

Consider the RSA system described on the previous page. The transmission band of interest is downconverted to an IF and then digitized. The time domain samples are digitally converted to a baseband record consisting of a series of I (in-phase) samples and Q (quadrature) samples. The DFT is performed sequentially on each segment of the IQ record, producing a mathematical representation of the occupied frequencies versus time, as shown in Figure 2-6 (page 16).

Obtaining an equally spaced sequential DFT is mathematically equivalent to passing the input signal through a bank of passband filters and then sampling the amplitude and phase at each filter output. The frequency domain behavior over time can be represented as a spectrum

The real-time DFT efficiently samples the entire incoming signal at the rate at which the new spectrum is calculated. Events that occur between the time segments where the FFT is performed are lost. The RSA minimizes or eliminates "dead time" by performing a hardware-based DFT, typically performing the transform on overlapping time segments at the fastest sampling rate.

In contrast, a swept spectrum analyzer is tuned to a single frequency at any given time. As the sweep progresses, the frequency changes, forming the diagonal line shown in Figure 2-7. As the sweep is slowed, the slope of the line becomes steeper so that the function of the spectrum analyzer in zero span can be represented as a vertical line, indicating that the instrument is tuned to a single frequency over time. Figure 2-7 also shows how sweeping can miss transient events, such as the single frequency jump in the figure.

RBW on a Real-Time Spectrum Analyzer

Frequency resolution is an important spectrum analyzer specification. When we try to measure signals with frequencies that are very close together, the frequency resolution determines the spectrum analyzer's ability to distinguish between these signals. On a traditional spectrum analyzer, the IF filter bandwidth determines the ability to resolve adjacent signals, also known as the resolution bandwidth (RBW). For example, to resolve two signals of equal amplitude but 100 kHz apart in frequency, the RBW must be less than 100 kHz.

For spectrum analyzers based on DFT technology, RWB is inversely proportional to the acquisition time. When the sampling frequency is the same, more samples are required to achieve a smaller RBW. In addition, the window function also affects the RBW.

Window Functions

In the discrete Fourier transform (DFT) analysis operation, an inherent assumption is that the data to be processed is a signal that repeats regularly in a single period. Figure 2-7 depicts a series of time domain samples. For example, when DFT processing is applied to frame 2 in Figure 2-8, periodic expansion is performed on the signal. Discontinuities generally occur between consecutive frames, as shown in Figure 2-9.

These artificial discontinuities generate spurious responses that were not present in the original signal, an effect that results in an inaccurate representation of the signal known as spectral leakage. Spectral leakage not only generates signals in the output that were not present in the input, but also degrades the ability to observe small signals in the presence of nearby large signals.

Tektronix real-time spectrum analyzers use windowing techniques to reduce the effects of spectral leakage. Before performing the DFT, the DFT frame is multiplied by a window function with the same length between samples. The window function is usually bell-shaped, which reduces or eliminates discontinuities at the end of the DFT frame.

The choice of window function depends on the frequency response characteristics, such as sidelobe level, equivalent noise bandwidth and amplitude error. The window shape also determines the effective RBW filter.

Like other spectrum analyzers, the RSA allows the user to select the RBW filter. The RSA also allows the user to select from a number of commonly used window types. Directly specifying the window shape increases flexibility and allows the user to optimize for specific measurements. For example, special attention should be paid to the spectral analysis of pulsed signals. If the pulse period is shorter than the window length, a uniform window (no window function) should be used to avoid the effect of de-emphasis at both ends of the DFT frame. For further information on this topic, refer to the Tektronix Primer "Understanding FFT Overlap Processing Techniques for Real-Time Spectrum Analyzers."

The frequency response of the window function determines the shape of the RBW. For example, the RWB on the RSA6000 is defined as the 3 dB bandwidth, which is related to the sampling frequency and number of samples in the DFT as follows:

Where k is the coefficient associated with the window, N is the number of time domain samples used in the DFT calculation, and Fs is the sampling frequency. For a Kaiser window with beta1=16.7, k is approximately 2.23. The RBW shape factor is defined as the frequency ratio between the 60 dB and 3 dB spectrum amplitudes, which is approximately 4:1. On the RSA6000, spectrum analysis measurements use Equation 2 to calculate the number of samples required for the DFT based on the input bandwidth and RBW setting.

Figures 2-10 and 2-11 show the time domain and spectrum of the Kaiser window used in the RSA6000 analysis. This is the default window used by the RSA6000 in spectrum analysis. Users can select other windows (such as Blackman-Harris, Uniform, Hanning) to meet special measurement requirements, and other windows can also be used when performing some of the measurements provided in the instrument.

Discrete Fourier Transform (DFT) in a real-time spectrum analyzer. DFT is defined as follows:

This is the basis of a real-time spectrum analyzer and is used to estimate individual frequency components x(k) from an input sequence x(n). The DFT is block-based, with N being the number of samples per DFT block (or frame). The input sequence x(n) is a sampled version of the input signal x(t). Although the input sequence is only defined for integer values ​​of n, the output is a continuous function of k, where k = (NW)/(2P) and W is the frequency in radians. The magnitude of X[k] represents the magnitude of the frequency component at frequency W that was present in the x(n) input sequence.

There are many efficient methods for calculating the DFT, such as the Fast Fourier Transform (FFT) and the Chirp-Z Transform (CZT). The choice of implementation depends on the specific needs of the application. For example, the CZT is more flexible than the FFT in terms of selecting the frequency range and number of output points. The FFT is less flexible but requires fewer calculations. Both the CZT and the FFT are used in RSA.

The ability to resolve frequency components depends not only on the specific DFT implementation, but also on the time length, or RBW, of the input sequence.

To illustrate the relationship between the DFT and FFT and CZT, we will analyze a sampled continuous wave (CW) signal. For clarity, we will use a real-valued sine wave x(t) as the input signal (Figure 2-12). The sampled version of x(t) is x(n). In this case, N = 16 and the sampling rate is 20 Hz.

Figure 2-13 shows the results of the DFT for 0 ≤ k < N. Note that for W > P (f > 10 Hz), the magnitude of X[k] is the mirror image of the upper half. This is the result for the real-valued input sequence x(n). In practice, the results for P< W < 2 P are discarded (or not calculated) when analyzing real input signals. For complex input, a unique result is obtained for 0 ≤ W < 2 P (0 ≤ f < 20 Hz).