# Spectrum Analyzer Determined by Choice of Measurements

## Overview

When performing spectral analysis on a high-frequency signal, the application determines the best instrument to use. Two principal types of spectrum analyzers exist, the Fast Fourier Transform (FFT) analyzer and the swept spectrum analyzer. This article presents the basic operation of both types of analyzers and discusses the advantages of each, giving the reader the necessary guidance to choose the right analyzer for their application.

## Fast Fourier Transform (FFT)

The Fast Fourier Transform (FFT) is based on the conversion of a time domain waveform to the frequency domain. Joseph Fourier developed the Fourier transform, which converted continuous time domain signals into continuous frequency domain information. The frequency domain information includes magnitude and phase values. However, the FFT analyzer samples discrete points over a certain time interval in the time domain and then records the points digitally. Because the time domain waveform is stored as discrete values, the waveform is not continuous and cannot be converted to the frequency domain using the standard Fourier transform. Instead, another version of the Fourier transform, the discrete Fourier transform (DFT), is used to convert the discrete time domain waveform into a discrete frequency domain spectrum.

Because the DFT uses discrete values to transform the time domain waveform into the frequency domain, the frequency domain is divided up into discrete frequency components, or "bins." The number of frequency bins over a given range, called the frequency resolution, is dependent on the number of discrete time domain samples taken. Thus, in order to boost frequency resolution, the number of samples must increase. A limiting factor here is execution time. The more samples that are taken to increase the resolution, the longer it takes to compute the DFT. The FFT is simply a faster method of calculating the DFT. It requires that the time domain waveform contain a number of samples equal to some power of two.

## FFT Spectrum Analyzer

The FFT analyzer consists of a number of components, each of which is necessary for frequency analysis. First, signals must pass through attenuators or gain stages to adjust the amplitude of the signal to match the input range so as to maximize the resolution of the signal. The signal is passed through an anti-aliasing filter, which we discuss in detail later. Next, a sampler takes discrete points of the analog waveform at a given frequency. This is followed by an analog-to-digital converter (ADC), which converts the samples into digital data. The data is transformed to the frequency domain using the FFT algorithm. The analyzer can use dedicated DSP circuitry to perform the FFT, or the data can be transferred to a PC where the FFT is calculated using software.

The rate at which points are taken by the sampler is a very important parameter of the FFT analyzer. In order for an analog waveform to be represented correctly with discrete values, it must be sampled at a sufficiently high rate. Otherwise the analyzer will interpret the input as a signal of frequency lower than the actual value. According to the sampling theorem, a signal must be sampled at a rate of twice the highest frequency component (the Nyquist rate). Any component whose frequency is higher than the Nyquist rate will appear in the measurement as a lower frequency component. This phenomenon is called aliasing. To avoid aliasing, an analog low-pass filter is placed at the input before the sampler. The low-pass filter determines the highest frequency of the FFT analyzer. Because the rate at which signals can be represented without error is one half the maximum sampling rate, signals are often cut off at a lower frequency to provide sampling rates greater than twice the maximum frequency components. Typically the cutoff of the low-pass filter is 2.5 times less than the maximum sampling rate of the analyzer. This determines the maximum frequency component.

The FFT algorithm is calculated from time records. Each record contains a finite number of discrete points. The FFT converts these time domain points into a finite number of frequency domain points. The frequency domain representation is symmetric about the 0 Hz frequency point. Only the upper half of the transform, which pertains to 0 Hz and above, is kept. Thus, only a certain number of frequencies can be analyzed. The frequency resolution, or spacing between the bins, must be such that the acquired frequencies are represented properly. In order to increase the frequency resolution the number of samples must increase. This is because the frequency resolution is inversely proportional to the length of the time record.

The transformed frequency domain data has an initial frequency value of 0 Hz and extends to 1/2 the sampling frequency, the maximum range of the FFT. If the number of samples is kept equal as the frequency range increases, the spacing between frequency bins increases. Thus the frequency resolution decreases. An alternative is to increase the number of samples acquired. However, increasing the number of samples requires a greater computational burden. In order to decrease the sampling rate without causing aliasing, a digital filter may be inserted between the ADC and the FFT to remove frequency components that are above twice the current sampling rate.

Because of the previously mentioned conditions, the measurement of higher frequencies with good resolution becomes quite difficult. However, by inserting a digital mixer between the ADC and the digital filter, a discrete sine wave can be multiplied to the incoming signal and effectively decrease the signal frequency. By reducing the signal frequency, the FFT can be performed on a range that has an offset from 0 Hz. This range is called band selectable analysis or zoom FFT analysis.

Whether an FFT analyzer is considered real time or not is determined by the computing time of the FFT. If the time to compute the FFT was faster than the rate at which the data was sampled, then an FFT could be performed for each point. However, this would require computational power beyond that of most instruments. A shortcut is to take a certain number of samples and transfer them to an intermediate buffer. The FFT takes the data from this buffer to perform its operation. If the FFT is computed before the next time record is placed in the buffer, then the analyzer is running in real time.

Because the number of samples in the time record is inversely proportional to the frequency span, it takes less time to compute the FFT for larger frequency spans. The point at which the frequency span time record equals the time it takes to compute the FFT is called the real-time bandwidth. To increase the speed at which the analyzer performs the FFT, dedicated DSP circuitry is placed on the instrument. Stand-alone and computer-based instruments use this method. Another method used by computer-based instruments is to use a digitizer to acquire the data, then transfer the data to a memory buffer. The FFT is then performed on the host PC in software.

## Swept Spectrum Analyzer

The swept spectrum is based on a different configuration than the FFT analyzer. It uses what is termed as a super-heterodyne configuration. Heterodyne refers to mixing and super refers to super-audio frequencies, or frequencies above the audio range. Similar to the FFT analyzer, the swept spectrum analyzer uses an attenuator and gain stages to adjust the signal to fit the input range of the analyzer. A voltage-controlled oscillator (VCO) sweeps through a range of frequencies that are mixed with the incoming signal. The signal from the input and the signal from the VCO are passed through a mixer. A mixer is a nonlinear device that produces the sum and difference of the original signal and the signal from the VCO, as well as the original signals and their harmonics. An intermediate frequency (IF) filter extracts the desired sum or difference of the original signals. The detector produces a voltage level relative to the amount of power received from the incoming signal. As the VCO sweeps different frequencies, the detector produces a corresponding voltage level, or power measurement.

An important parameter of the swept analyzer is resolution bandwidth. It is determined by the IF filter portion below the minimum insertion-loss point. Often the IF filter section is composed of a number of filters, which together determine the resolution bandwidth. Resolution bandwidth is affected by the speed at which the instrument sweeps the frequency range. The IF filter requires a certain time to respond to signals placed at the input, so the analyzer cannot sweep through the frequency range too fast. Two errors can result from sweeping through the frequency range too quickly. First, the amplitude is at a lower level than if it were swept at a slower speed. Second, the signal shifts up in frequency.

Because any time involved in sweeping the frequency range causes errors in the signal due to the filter response time, the instrument should sweep infinitely slow. However, this is not practical, so an acceptable error must be determined from which the sweep rate can be calculated. The following equation is used to calculate the maximum sweep rate according to the resolution bandwidth (RBW). where k is a constant that is dependent on the resolution bandwidth of the filter. The sweep rate is proportional to the resolution bandwidth squared.

## Comparison

One of the principal advantages that the swept analyzer has over the FFT analyzer is its frequency range. Because the FFT analyzer needs to acquire a digital time domain waveform, the frequency range of the FFT analyzer is dependent on the sampling rate of the ADC. The sampling rate of the analyzer determines the highest frequency that can be analyzed with the FFT. Current ADC technology has brought sampling rates into the MHz range. Swept spectrum analyzers, on the other hand, can measure frequencies in the high GHz range.

A key advantage of the FFT analyzer over the swept spectrum analyzer is its speed. Because the sweep rate is proportional to the resolution bandwidth squared, the sweep rate is decreased for smaller resolution bandwidths. This increases the total measurement time. The FFT analyzer is only limited in time by how long it takes to acquire the data and to compute the FFT.

In order to analyze signals that change in frequency over time, it is important to acquire data fast enough to view the changes in frequency. An example of this type of signal is with transient analysis. Swept spectrum analyzers are only capable of detecting continuous wave (CW) signals. The length of time it takes to complete a frequency sweep on the swept spectrum analyzer makes it inappropriate for measuring fast changing signals at lower frequencies. The FFT analyzer is much faster than the swept spectrum analyzer in acquiring spectrums, especially if the analyzer has dedicated DSP to perform the FFT. If the FFT analyzer is a real-time system, then you can measure the signall without losing any of the frequency data over time. For this reason, the FFT analyzer is a better choice for acquiring signals that change with frequency.

If phase information is desired, you must use the FFT analyzer. The FFT analyzer is based on the Fourier algorithm, which includes both real and imaginary numbers. From these numbers, you can calculate the magnitude and phase at each frequency. The swept spectrum analyzer can only acquire magnitude information for the user.