**Time as well as frequency measurements describe digital modulation. Modern signal analyzers provide both.**

A few decades ago, when analog oscilloscopes and spectrum analyzers were state-of-the-art instruments, an engineer's life was far simpler than today. Of course, it also was much less flexible and many would say less interesting. An RF engineer didn't worry about DC drift, and a DC voltmeter designer didn't care about voltage standing wave ratio (VSWR).

The frequency and time domains were separate. RF engineers measured power at certain frequencies, and baseband circuit designers measured rise times and pulse widths. Pulse width modulation (PWM) bridged the time-frequency divide in a simplistic manner by using the duty cycle of a pulsed RF signal to adjust a continuous wave (CW) power reading.

Recently, signal modulation has become much more complex, and the exact time that a particular frequency and phase are transmitted matters. Communications systems are using digital signal processing (DSP) to combine fast frequency hopping with simultaneous phase and amplitude modulation. A traditional swept-frequency spectrum analyzer can easily ignore power at one frequency while tuned to another. Rapidly changing signals demand new types of test instruments.

You can t infer much about the nature of digital modulation from the RF analog circuitry following the input of a modern microwave spectrum analyzer. Up to the intermediate frequency (IF) stage, the design goal is the same as in conventional swept analyzers: to recover the relatively low-frequency modulation from the much higher frequency carrier without adding significant impairments. It is true that the trend toward very wide modulation bandwidths complicates the design, but the RF circuitry remains analog. In contrast, digital IF stages now are the norm (**Figure 1**).

**The Digital IF**

In communications systems that only operate with narrowband signals, the IF can be under'sampled by a limited sample rate ADC with a large bandwidth. Nyquist's criterion that the sampling rate be twice the signal bandwidth (not the carrier bandwidth) can be applied with care to the modulation bandwidth at the IF stage. Several 10-MS/s to 20-MS/s ADCs optimized to have very good spurious-free dynamic range (SFDR) at 70-MHz or 80-MHz carrier frequency have been developed. However, undersampling provides the greatest benefit when the ratio of IF carrier frequency to modulation bandwidth is large.

A measuring instrument, as opposed to a specific communications system, must provide a general-purpose solution, and modulation bandwidths of at least 60 MHz are becoming routine in some application areas. To deal with this wide bandwidth, a high sampling rate ADC is used.

Today, ADCs are available with the necessary sampling speed and bandwidth to directly oversample a wide analysis bandwidth at greater than Nyquist and with at least 14-b resolution. This means that whatever signal bandwidth has been accommodated through the RF downconversion and anti-alias filtering ahead of the ADC, it will be preserved in the digital output.

*Aliasing*

Anti-alias filtering ahead of the ADC is required because of the spectrum folding that occurs in sampled data systems. If signal components are present with frequencies above f_{S}/2 where f_{S} is the sampling rate, they will fold back into the baseband spectrum and corrupt it.

Once aliasing has occurred, it cannot be corrected. An anti-alias filter must trade off good pulse response against the cost in additional bandwidth that a linear phase filter may require. Linear phase across all frequencies of interest corresponds to a constant group delay of signal components in the time domain.

A practical high-order anti-alias filter may have a transition band about 0.1 f_{S} wide, extending from 0.4 f_{S} to f_{S}/2. This means that if you need a 10-MHz passband, you must sample at a minimum of (0.5/0.4) • 2 • 10 = 25 MS/s. The factor of 2 satisfies the Nyquist criterion, and the ratio 0.5/0.4 accounts for the excess bandwidth required by the filter's transition region. Your sampling rate must support a theoretical 12.5-MHz bandwidth because 2.5 MHz is used by the filter transition region.^{1}

As shown in Figure 1, the data output from the ADC represents successive time-ordered samples of the anti-alias filtered IF signal. Because the samples preserve the full modulation bandwidth, all the relevant characteristics of the IF signal have been captured in digital form. Typically, a data processing frame might comprise 1,024 contiguous samples.

*FFT*

If all you needed was the spectrum corresponding to the modulation, the frame of digitized data could be fed to a DSP chip to perform an FFT. As is well known, the FFT algorithm operates on a block of data to produce the corresponding frequency-domain spectrum.

The time relationship among the various frequencies is lost in the spectrum an FFT returns. However, if successive frames of contiguous data are operated on, time resolution equal to the length of a frame is achieved.

_{S}/2, and the frequency bin resolution is f

_{S}/N where N equals the number of samples in the time record. For a 60-MHz modulation bandwidth and assuming that 25% additional bandwidth is required because of the anti-aliasing filter, you need a 150-MHz ADC sampling rate.

For a typical 1,024-sample time record, the corresponding frequency-domain spectrum would comprise 513 lines starting at zero (DC), extending to 75 MHz, and spaced 146.5 kHz apart. In other words, the resolution of the FFT is 146.5 kHz, which is much too coarse to provide any meaningful detail within a smaller part of the overall frequency band.

To concentrate on a smaller bandwidth, or span, you need to perform a zoom FFT. The frequency band of interest must be selected, downconverted to baseband, resampled, and filtered before the FFT is performed. For example, if the frequency band from 20 MHz to 30 MHz were of interest, then the time-domain data output from a 20-MHz to 30-MHz digital bandpass filter would be multiplied by a digital 25-MHz sine wave to produce baseband data either side of DC.

You want a 10-MHz overall bandwidth in the resulting FFT, so the sample rate must be reduced from 150 MHz to 20 MHz. This is accomplished by a combination of resampling that includes interpolation and decimation and filtering to restrict the bandwidth. After the DC-to-10-MHz FFT is computed, the displayed values are relabeled to correspond to the 20-MHz to 30-MHz bandwidth. If 1,024 resampled data points are used, the resulting 513-line FFT will have 19.5-kHz resolution.^{2}

The characteristics of the analog anti-alias filter ahead of the ADC never have to be altered because the ADC runs at a constant sampling rate. Digital filtering removes aliases caused by the resampling process and is performed by a DSP. This means that no hardware changes are needed to cater for whatever span and resolution you may have selected.

Because the digital modulation stand-ards used in modern communications systems are defined in terms of in-phase (I) and quadrature (Q) orthogonal components, vector signal analyzers (VSAs) and spectrum analyzers with digital modulation analysis capabilities combine quadrature detection with digital downconversion ahead of the FFT stage. This means that two multipliers are used, one driven by the center frequency sine wave and the other by the cosine wave.

In practice, digital downconversion (DDC) may begin by digitally filtering the IF output to eliminate all but the frequencies in the band of interest. The actual frequency conversion involves a numerically controlled oscillator (NCO) tuned to a multiple of the desired center frequency and used to address sine and cosine lookup tables. The resulting synthesized sine (Q) and cosine (I) waveforms then are used to digitally multiply the corresponding band-limited input data.

The IF downconversion process uses a mixer, which really is a multiplier. Mixers are based on the trigonometric identity 2(CosA)(CosB) = Cos(A+B) + Cos(A-B). Ideally, multiplying one sine wave by another produces half-amplitude sine waves at the sum and difference of the original signal frequencies.

In the RF stages ahead of the digital IF, one input to a mixer is from a local oscillator (LO) and the other from a preceding downconversion stage or the signal input. Because RF mixers are analog circuits, they are not perfect, and additional harmonics and their products also appear at the mixer output.

In general, a mixer produces IF frequencies f_{IF} = m • f_{RF} n • f_{LO} where m and n indicate harmonic orders. For example, if n and m are 1, the expected sum and difference IF frequencies are produced. Values greater than 1 correspond to the second harmonic of the LO mixing with the third harmonic of the input RF, the RF input second harmonic mixing with the LO third harmonic, and so forth.^{3}

In the case of a zoom FFT with digital IF data, there are no nonlinearities to worry about. A digital multiplier simply multiplies. Of course, multiplying two k-bit numbers results in a product with 2k bits of resolution, so truncation can cause loss of accuracy.

Following the digital downconversion stage, filtering is necessary to reject the unwanted f_{IF} + f_{LO} frequencies. The remaining f_{IF} – f_{LO} signal in this case runs from -5 MHz to +5 MHz and directly corresponds to the 20-MHz to 30-MHz section of the overall frequency response we were interested in. The circuitry from the ADC through to the beginning of the data analysis sections is called a digital IF, corresponding to the functions traditionally accomplished in an analog IF section.

Usually, the last IF filter stage sets a spectrum analyzer's resolution bandwidth (RBW). If two signals are closer than the RBW, especially if one is much larger than the other, they will not be displayed as distinct peaks. For small RBW values, digital IFs are several orders of magnitude faster than conventional swept analyzers.

**Further Analysis and Time Relationships**

Whether the IF data is stored to a large memory or analyzed in real time depends on how quickly it can be processed. Spectrum analyzers with digital IFs can store digitized time-domain values for later analysis. A few can analyze contiguous blocks or frames of data quickly enough to have no gap between samples.

FFTs contain no time reference so the timing aspect of a spectrogram, for example, is provided by the frame-to-frame time. Unfortunately, another characteristic of FFTs, the need for windowing, may affect the detection of very fast transient signals.

Typically, windowing functions reduce the amplitude of the samples within a frame to zero at each end of the frame. This is necessary in general because signals are not sampled coherently: There is not an integer number of cycles within a time frame. Without windowing, spectral leakage occurs, which smears the power associated with one frequency across adjacent FFT bins, giving an incorrect representation of the signal.

But if a fast transient happens to occur at the edge of a frame, it will be lost. Overlapped FFTs can alleviate this situation, but interpreting the displayed spectrum may take some care. This type of analysis is performed by adding only a fraction of a complete time frame of data samples to successive FFTs.

In other words, after the first 1,024 points have been acquired and the FFT computed, the next FFT might be calculated for the most recent 1,000 points of that frame plus 24 new points. Computing successive FFTs in this manner eventually will bring a fast transient into the center of the window and allow its frequency components to be extracted. So, the method acts to magnify time resolution although the frequency-domain display can be confusing.^{4}

In analyzers that go beyond frequency-domain presentations, timing is especially important. Demodulation begins by building a dynamic model of the ideal modulation that would occur for a known communications format. The ideal is compared with the actual received data and the errors computed.

This process supports computation of the error vector magnitude (EVM) metric and the use of constellation diagrams to show ideal vs. actual phase/amplitude states. Instruments capable of such analysis may be called VSAs or simply spectrum analyzers with vector signal analysis capabilities.

**Real Instruments**

*Anritsu*

Anritsu's Model MS2781A Signature Signal Analyzer can perform real-time signal capture into memory for an extended but not infinite period of time. When the type of signal being measured is known, a large finite amount of data usually is sufficient. Most modulation measurements require a few hundred or thousand symbols, so a memory length of 10 MS or 1.28 s is more than adequate to capture I/Q vectors.

Many startup transients are in the millisecond range. Some drift measurements require a longer time, but the instrument can be set to acquire data for several seconds. Alternatively, bursts of samples can be acquired periodically and recorded to disk. MATLAB running from the Windows-based analyzer can be used to extract virtually any desired information from the stored I/Q vectors, such as FFTs, spectrograms, frequency vs. time plots, global system for mobile communications (GSM) demodulation, and wideband zero-span displays.

As in any RF analyzer, the MS2781A's analog specifications determine how well the input signal will be captured. In designing the Signature model, Anritsu has stressed a low noise figure, phase noise, and residual EVM over the instrument's 100-Hz to 8-GHz frequency range. With the instrument's 30-MHz capture bandwidth, 50 MHz without anti-aliasing filtering, you can acquire one Bluetooth hop or a wideband code division multiple access (WCDMA) channel and its upper and lower adjacent and alternate channels five in total.

*Agilent Technologies*

The PSA Series Spectrum Analyzer is a swept-tuned instrument with certain VSA capabilities. The completely digital IF supports a synthesized swept tuned mode or an FFT mode of operation. Analysis bandwidth of 40 MHz or 80 MHz is optional. Real-time corrections are applied on the fly to the ADC samples to calibrate the output I/Q data for optimal flatness and dynamic range.

If the digitized signal is analyzed immediately and allowed to run continuously, gaps eventually will appear because data is entering the system faster than it can be processed and displayed. However, if the data is recorded to the PSA's large memory, from 1.28 s at an 80-MHz bandwidth up to hours of data at a slower speed can be analyzed off-line. For example, Agilent's 89600 VSA software is appropriate for this purpose.

Interestingly, the data can be played back at an adjustable speed so that short-term events won t be missed. Resampling within the PSA analyzer can be set to an arbitrary rate with up to 10 digits of resolution.

Triggering determines what data the analyzer will record. The PSA supports a free-running mode, external event triggering, IF envelope, and frame triggering. Fast time-domain averaging based on frame triggering can reduce the noise floor in repetitive signals by 20 dB or more and is faster than traditional trace-to-trace averaging by several orders of magnitude.

*Tektronix*

Recently, Tektronix introduced the Model RSA3408A Real-Tme Spectrum Analyzer (RTSA) with 36-MHz bandwidth provided via a 102.4-MHz 14-b ADC and up to 65.6-MS acquisition memory size. As with the Agilent and Anritsu products, the RSA3408A can capture 1.28 s of data at its fastest sampling rate, but the RSA3408A also provides continuous acquisition and display features.

The frequency range is DC to 8 GHz with RBW selections from 1 Hz to 100 kHz and several types of available FFT windowing functions. However, beyond the signal capture and processing specifications, the analyzer has distinctive triggering capabilities.

In the RTSA, signal analysis is performed on the digitized data before it is stored in memory as shown in **Figure 2.** The processing keeps up with the data rate and enables frequency-selective triggering. As in a DSO, the acquisition memory is arranged to allow continuous circular storage. When the trigger conditions are satisfied, an amount of data ahead of the trigger can be retained and the acquisition continued from that point to fill the memory.

In this way, the RTSA supports very selective signal capture based on precise criteria. Because this approach allows long-term monitoring and acquisition by exception, it is similar to the triggering facilities on logic analyzers and DSOs. Of course, more general triggering modes also are available and aid troubleshooting less well-defined situations.

**Summary**

In describing some of the ways in which time-domain and frequency-domain functionality have come to reside in a common instrument enclosure, many details have been glossed over. For example, in addition to modern spectrum analyzers having a digital IF and perhaps vector demodulation capabilities, the instruments are extremely complex.

To provide accurate signal analysis, you first need to develop a very good spectrum-analyzer RF section. Partly, this is accomplished through the use of new components and enhanced RF circuit design and simulation capabilities. In addition, the DSPs that provide so much of the signal-processing horsepower also run correction algorithms that ensure accurate calibration under all conditions.

Contributing to the analyzer's use model, also courtesy of DSP processing, is the capability to mimic the simpler swept analyzer. Swept spectrum analyzer users know what result they should see for a certain test. Although a modern digital machine may have many advanced modes of operation, it generally can operate like a swept analyzer should you so wish.

Before you can decide which kind or brand of signal analyzer you may need, you must understand the time/frequency nature of the signals involved in your work. If you design or test communications systems, it's unlikely that only a time or frequency capability will suffice: You need both. The question is how much of each and in what form.

**References**

1. *Agilent Vector Signal Analysis Basics,* Application Note 150-15, Agilent Technologies, 2004.

2. Bores, C., The zoom• FFT,• Bores Signal Processing,http://www.bores.com

3. *Dynamic Performance Requirements for High-Performance ADCs and RF Components in Digital Receiver Applications,* Application Note 3062, Maxim/Dallas Semiconductor, February 2004.

4. *Understanding FFT Overlap Processing, A Tektronix Real-Time Spectrum Analyzer Primer,* Tektronix, 2005.

**FOR MORE INFORMATION**

*on the MS2781A Signature Signal Analyzer*

www.rsleads.com/510ee-179

*on the PSA Series Spectrum Analyzer*

www.rsleads.com/510ee-180

*on the RSA3408A Real-Time Spectrum Analyzer*

www.rsleads.com/510ee-181