#### Seeing Frequency Components

Examining the frequency components of a waveform can reveal important information that may not be obvious on a typical oscilloscope display. For example, waveform distortion or symmetry problems may not be apparent from a standard waveform display (Figure 1). Yet those problems can become clearly obvious with a simple look at the waveform's frequency components (Figure 2).

*Figure 1. This sinewave, which appears normal on a typical oscilloscope display, is actually a distorted sinewave. The distortion becomes apparent in the TDS 300 Series FFT display shown in Figure 2.*

*Figure 2. The TDS 300 Series FFT display (bottom) of the seemingly normal sinewave in Figure 1 reveals that there is distortion. The distortion is indicated by the presence of harmonics other than the sinewave's fundamental frequency. The measurement cursor has been placed on the fundamental (2 MHz) in this display. Note the third harmonic at 6 MHz.*

In the past, viewing a waveform's frequency components required a spectrum analyzer and the additional skills for its use. That's still the case for some types of in-depth frequency analysis. However, many types of basic frequency analysis can now be done with a digital storage oscilloscope (DSO) such as the Tektronix TDS 300 Series.

To provide the added dimension of viewing waveform frequency components, the Tektronix TDS 300 Series has a built-in FFT (Fast Fourier Transform) capability. In essence, the FFT displays a waveform's frequency components. This application note covers the basics of TDS 300 Series FFT frequency displays, what they mean, and how to use them.

#### The Basic Building Blocks of Waveforms

Understanding FFT frequency displays starts with first understanding waveforms and their basic building blocks. This includes making the distinction between periodic waveforms and nonperiodic waveforms. The easiest place to begin is with periodic waveforms.

**Basics of Periodic Waveforms.** Periodic waveforms are waveforms that repeat over regular intervals or periods of time. Sinewaves, squarewaves, and triangular waves are common examples of periodic waveforms.

According to Fourier theory, all periodic waveforms are made up exclusively of specific sinusoids. The base sinusoid is referred to as the fundamental. Its frequency is the same as the waveform's frequency. For example, a 1 kHz square wave has a fundamental sinusoid that also has a frequency of 1 kHz. Similarly, a 1 kHz triangular wave has a fundamental of 1 kHz. In essence, the fundamental is the dominant frequency component of the waveform. As such, it determines the waveform's frequency or period of repetition.

Along with the fundamental component, all nonsinusoidal periodic waveforms also have harmonic components. A harmonic is a sinusoid with a frequency that is an integer multiple of the fundamental. For example, the third harmonic of a 1 kHz square wave is a 3 kHz sinusoid, the fifth harmonic is a 5 kHz sinusoid, and so on out to infinity.

In addition to having specific frequencies, a periodic waveform's fundamental and harmonics also have specific amplitude and phase relationships. These relationships allow the fundamental and harmonics to add up to the specific waveform. This is illustrated further in Figure 3, where the first five frequency components of a squarewave are summed.

*Figure 3. All periodic waveforms are the sum of a fundamental sinusoid and specific harmonics. In this case, the fundamental, 3rd, 5th, 7th, and 9th harmonics are added in an attempt to obtain a squarewave. To get closer to the actual squarewave, additional higher harmonics are needed.*

Notice that the sum waveform in Figure 3 is not an exact squarewave. This is because not enough harmonics have been added. With additional higher harmonics, the waveform's transitions would become steeper, the corners squarer, and the tops and bottoms flatter.

In theory, all harmonics out to infinity are necessary for an ideal squarewave or any other nonsinusoidal wave shape. In reality, all waveforms are bandwidth limited, which means that their higher frequency components have been attenuated significantly.

Usually, a waveform's significant frequency components are illustrated with spectral lines. This is shown in Figure 4, which depicts the frequency spectrum of a squarewave. In this diagram, the frequency components are represented by vertical lines with each component placed at its frequency on a frequency axis. The length of each spectral line indicates the frequency component's amplitude relative to other components.

The basic goal of most frequency analysis methods is to obtain a spectral diagram or display similar to Figure 4. Various methods can be used to reach this goal. In spectrum analyzers, swept filters are used to detect the individual frequency components of a waveform. With DSOs, such as the Tektronix TDS 300 Series, spectral displays are obtained by digitally analyzing the waveform with a built-in Fast Fourier Transform (FFT) program.

**Basics of Nonperiodic Waveforms.** Nonperiodic waveforms do not repeat themselves. Unlike periodic waveforms, they do not have a regular interval (period) of repetition.

The single pulse shown in the upper display in Figure 5 is an example of a nonperiodic waveform. Since it is but one pulse, it does not repeat itself. Thus, it has no period and is nonperiodic.

*Figure 5. A single pulse (upper) is an example of a nonperiodic waveform. It was captured using the TDS 300 Series single-acquisition mode with the FFT function selected. The FFT display (lower) for such single-event waveforms is a continuous frequency spectrum rather than the discrete frequency lines that distinguish periodic waveforms.*

The lower display in Figure 5 shows the FFT display for the single pulse. Notice that, rather than being individual spectral lines, the display is a continuum, or envelope, of frequency amplitudes. This continuous spectrum is a distinguishing feature of single-event (transient) waveforms. Transients contain all frequencies not just harmonics and have some regular variation in spectrum amplitude. In the case of the square pulse, the frequency amplitude variation follows a (sin x)/x envelope. Other pulse shapes will have other characteristic spectrum shapes.

Some examples of transient waveforms are static discharges, contact arcing, and switching transients. These frequency-rich waveforms often make their presence known through interference with other systems. A common case is the static burst caused on radios and TV sets by a distant lightning storm. Another example is power-line spikes and other interference that can occur when machines, such as an air conditioner, switch on or off.

While transients are common examples of nonperiodic waveforms, they are not the only examples. A nonperiodic waveform can also be continuous or ongoing, as opposed to being a single, short-lived transient pulse.

Data streams, mechanical vibration waveforms, and random noise are typical examples of nonperiodic waveforms. Like single-event transients, such waveforms should be acquired and stored with single-shot triggering. Single-shot acquisition will provide a stable, snap-shot view of the waveform for measurement and analysis. With the Tektronix TDS 300 Series, the analysis can include examining the waveform's frequency content with the FFT display.

#### Obtaining an FFT Display

Tektronix TDS 300 Series DSOs use a mixed-radix FFT routine to analyze waveforms for frequency content. The mixed-radix FFT has the advantage of providing results in a base-10 record length. This allows direct matching of frequency scaling to the base-10, 1-2-5 sequence of the DSO's time-base controls. This, in turn, allows easier interpretation of frequency results than would be the case with more common binary-based FFT record lengths.

To obtain an FFT display with the TDS 300 Series, follows these basic steps:

Performing the above steps results in an FFT display for the acquired waveform. The scaling for the FFT display is provided at the bottom of the TDS 300 Series screen and to the right of the MATH icon. Horizontal scaling (frequency) is in Hz/division with 0 Hz at the left edge of the display. Vertical scaling (relative amplitude) is in dBVRMS, which is dB relative to 1 VRMS (0 dB = 1 VRMS). Typically, these scale factor readouts are used as visual references only since far more precise waveform measurements can be made quickly and easily using the TDS 300 Series measurement cursor system.

The following items provide further important information regarding the TDS 300 Series FFT and its frequency displays:

**Signal Acquisition.**The FFT function can be applied to signals acquired with single-shot triggering (nonperiodic signals) or with repetitive triggering (periodic signals). Periodic signals should be acquired with the Signal Averaging function activated to reduce the effects of signal noise. A minimum of 16 averages is recommended.**Hanning Window.**Acquired signals are always multiplied by a Hanning Window function prior to performing the FFT. Details on this window and its application are provided in the TDS 300 Series Users Manual. For best results on pulse-type waveforms, use trigger positioning to acquire the pulse so that its peak is horizontally centered in the TDS 300 Series display, as shown in Figure 5.**Frequency Resolution.**There are 1000 frequency points in the displayed FFT results. These points are equally spaced from 0 Hz to the Nyquist frequency (FN) at the right side of the display. The Nyquist frequency is equal to one-half of the sample rate used in capturing the waveform and is 1/2 * 100/(time/div setting) when Fit to Screen is used. With Fit to Screen turned off, FN = 1/2 * 50/(time/div setting).

#### Identifying Waveform Frequency Components

The most basic use of the TDS 300 Series FFT display is to reveal the frequency components of a periodic waveform. This is sometimes called harmonic analysis since the frequency components of a periodic waveform are all harmonically related.

By acquiring any periodic waveform and applying the FFT function, you can obtain a display of the waveform's frequency components. Further information about the frequency components can be obtained by using the TDS 300 Series measurement cursors. Simply place the cursors on the components of interest, and the display readout shows the corresponding frequency and amplitude values.

Nonperiodic waveforms can be analyzed in a similar manner. However, remember that a transient pulse has a continuous spectrum of frequencies instead of discrete harmonics. For such continuous spectra, analysis usually focuses on main-lobe 3 dB width and the relative amplitudes of side lobes (see Figure 6).

#### Identifying Waveform Distortions

Another basic use of the TDS 300 Series FFT display is to identify waveform distortions. This is illustrated in Figures 1 and 2, which show a "normal looking" sinusoid and its corresponding frequency view.

Notice that the sinusoid in Figure 1 looks normal. However, the TDS 300 Series FFT display in Figure 2 shows that there are some low-level frequency components in the sinusoid's frequency content. These additional low-level components indicate that the sinusoid is not pure and that it has harmonic distortion. If the sinusoid were pure, its frequency content would consist of a single spectral line at the sinusoid's frequency.

Most real-life sinusoids do have some amount of harmonic distortion. The degree of this distortion is often described by specifying the level of certain harmonics relative to the sinusoid's fundamental frequency. For example, third harmonic distortion might be specified as being 50 dB down from the fundamental. This can be checked by using the TDS 300 Series FFT display and measurement cursors to identify and measure the different harmonics and their levels (Figure 2).

Waveform symmetry can also be verified by using the TDS 300 Series FFT display. For example, a squarewave with symmetric top and bottom squares should be composed of the fundamental frequency and odd harmonics only. If the squarewave is not symmetric, even harmonics will also appear in the FFT display. The levels of these even harmonics will increase with increasing squarewave nonsymmetry.

Caution should always be used in identifying harmonics in FFT displays. This is because sometimes apparent harmonic components are actually aliases of higher frequency components. This is illustrated in Figure 7, which shows a squarewave and its FFT display. At first glance, it appears that there are significant even harmonic components residing between the higher odd harmonics. This actually isn't the case. What has occurred here is that higher frequency odd harmonics have folded down into the display to appear as even harmonics. The clue is that the harmonics have not died down to an insignificant level at the right side of the display (the Nyquist frequency). Consequently, those higher frequency harmonics that would have been off screen to the right have folded around the right edge of the display to appear as low-frequency aliases. To minimize such aliasing, the squarewave should be re-acquired with fewer cycles in the time display. (For a detailed discussion of aliasing and other FFT characteristics, refer to ** The FFT: Fundamentals and Concepts**, Robert W. Ramirez, Simon & Schuster, 1985.)

#### Other Applications

The preceding examples have dealt primarily with harmonic analysis. The harmonics of the captured waveform have been examined to determine something about the waveform.

There are numerous other possibilities for using the frequency view to examine both periodic and nonperiodic waveforms. Briefly, here are a few of those examples:

**Amplitude Modulation (AM).** The FFT display of an AM signal shows the relative levels of the carrier frequency and the modulation sidebands. This is most apparent when a single-frequency test tone is used to modulate the carrier.

**Vibration Analysis.** By using a transducer, mechanical vibrations can be viewed on the TDS 300 Series as a vibration waveform. This overall vibration signature waveform is composed of vibration components from various elements of the mechanical system.

The individual vibration components can be viewed and analyzed with the FFT display. For example, signature analysis can be used to identify excessive roller bearing vibration within a system. The benefit of such analysis is that the deteriorating bearings can be detected and replaced before they seize and cause extensive machine damage and down time.

**Frequency Response.** The FFT can be used to obtain a quick estimate of a circuit's frequency response. This can be done by applying a narrow pulse (an impulse) to the input of the circuit and acquiring the impulse response waveform at the circuit's output. The FFT of this impulse response provides an estimate of the circuit's frequency response. Using the measurement cursors, this frequency response can be analyzed further for such information as bandwidth, roll-off, and out-of-band responses.

#### Some Tricks for Better Frequency Displays

What is seen in the frequency display depends on a variety of factors. A key factor, of course, is the waveform that is being transformed to the frequency display by the FFT. Other factors include the various artifacts of the waveform, such as its DC component, or offset, and noise present on the waveform. How the waveform is captured also has some bearing on what is seen in the frequency display.

The following methods can be used to address some of the key issues in obtaining better frequency displays:

**Remove the DC Component.** Some waveforms have a DC component or offset. In the frequency display, a large DC component can overshadow other frequency components. To minimize the DC component, acquire the waveform with AC coupling.

Even with AC coupling, a waveform may still appear to have a small, residual DC component. This is a result of capturing a non-integer number of waveform cycles in the display (e.g., 4.5 cycles versus 4 cycles). The easiest thing to do is to simply ignore the small residual DC component. Another approach is to acquire more cycles of the waveform, thus reducing the percentage contribution of any fractional cycle at the display edge.

**Acquire More of the Waveform.** For periodic waveforms, acquiring more cycles in the time display results in greater frequency resolution in the frequency display (see Figure 2). Typically, a minimum of four to eight cycles of a waveform in the time display results in the best use of the full frequency display.

When dealing with pulse or transient waveforms, be sure the entire pulse is on-screen, including some baseline prior to the pulse and some baseline after the pulse. For best frequency viewing results, the entire pulse should cover about 1/8 to 1/4 of the time display and should be centered on the display.

Be careful, though. If some of the higher-frequency components appear to be close to the right edge of the frequency display, aliasing errors may be occurring (see Figure 7). Aliasing is where high-frequency components beyond the edge of the display are folded back into the display as low-frequency components. If aliasing appears to be a problem, acquire the waveform so that it is spread out more over the time display. For periodic waveforms, this means acquiring fewer cycles. For transients, this means spreading out the pulse over more of the time display. With a little experience, you'll soon be able to judge how to acquire signals for optimum FFT displays.

*Figure 7. An example of aliasing or foldover where high-frequency components appear as low frequency components. When an FFT display does not decrease to near zero at the right side of the display, the remaining components fold back into the display as frequency aliases.*

**Signal Average.** Noise appearing on the captured waveform also appears as noise in the frequency display. In the case of random noise on periodic signals, the noise amplitude on the acquired signal can be reduced by signal averaging. The greater the number averages, the greater the reduction of noise in both the time and frequency displays.

Signal averaging is applied to periodic waveform acquisition by selecting **Average** in the TDS 300 Series **Acquire** menu. A minimum of 16 averages is recommended. This should provide clean time and frequency displays for typical periodic signals. When noisier signals are being captured, you'll need to select a higher number of averages for greater noise reduction.

As a final note, signal averaging can only be applied to periodic waveforms. This is because multiple acquisitions of the waveform must be taken in order to form an average. Since nonperiodic waveforms can only be acquired once, in a single-shot mode, they cannot be averaged.

#### Conclusion

The ability to see a waveform's frequency components gives you another dimension of knowledge about that waveform. In the past, obtaining a frequency display required using a spectrum analyzer or a laboratory bench quality DSO with FFT capability. That is no longer the case.

Today, you can take basic FFT capability into the field with the portability of the Tektronix TDS 300 Series DSOs. This gives you the added capability of seeing a waveform's frequency display. Being able to quickly check a waveform's frequency content and the shape of its spectrum is a powerful research and analysis tool. It's also an excellent troubleshooting aid for verifying circuit response, checking waveform distortion, and identifying and tracking down noise and interference problems.

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