Key Relationships Between Time and Frequency Domains

By Gary Breed Editorial Director

ith today’s com- • Oscillator startup behavior The observation and plex modulation • Clipping or limiting of signals analysis of signals in both Wformats, digital • Waveform distortion (usually to determine time domain and frequen- IF and baseband signal cause of harmonics or spurs) cy domain is especially processing, frequency • Analysis and measurement of modulation important today, given synthesis and control sys- envelope properties recent developments in tems, the relationship • On/off switching transition behavior digital signal processing between the time domain • Transients (DC power shifts, effects from and complex modulation and frequency domain other portions of the circuit) has become an essential • Power supply regulation effects concept for all engineers to understand. This • Noise analysis tutorial will present some of the basic relationships between these two perspectives. The Table 1 · Some common time domain emphasis is on identifying what factors are observations of RF signals. important and their effects. • Electromagnetic compatibility (EMC) mea- Signals in Time and Frequency surements and analysis Events observed only in time are sequen- • Signal integrity analysis (verify frequency tial (e.g. the rise and fall of a sine wave), while range of Fourier coefficients/harmonics) in the frequency domain, we are looking at • Analyze jitter as phase noise cyclical events (e.g. the same sine wave • Verify performance of digital filters and appears as a single frequency). In other words, other digital signal processing to observe the signal in the time domain, you • Analyze performance of digital-to-analog use an oscilloscope. For frequency domain converters (DACs) in direct digital fre- observations, you use a spectrum analyzer. quency synthesis (DDS) and other wave- Table 1 lists some of the most common form generation applications time domain measurements, observations and analyses of RF signals, while Table 2 lists the Table 2 · Some common frequency domain common observations of digital signals in the observations of digital signals. frequency domain. These represent the most basic elements that “crossover” between the two observational domains, where the oppo- mathematics, with the basic form: site perspective is used to evaluate the behavior in the normal domain. As we all learned in our undergraduate Linear Systems class, there are much more detailed mathematical relationships between where F(y) is is the frequency domain repre- the time and frequency domains. The Fourier sentation of the time domain function, f(x). transform is the foundation of time/frequency The above equation is for an impulse func-

54 High Frequency Electronics High Frequency Design TIME & FREQUENCY tion; a single event in the time domain. While impulse response is important, we are more often interested in periodic waveforms (signals that are present in the fre- t quency domain), for which the Fourier transform takes the form of a series: t

(a)

which can be reduced to:

where,

f(x) is a periodic function with a period 2p, that is, its

values are completely defined by an interval from 0 to (b) 2p radians (or –p/2 to +p/2). Figure 1 · Time domain (a) and frequency domain (b)

A0 is the initial value (analogous to a DC offset). representations of a waveform comprising a series of short pulses.

An, Bn are the coefficients of each term of the series

The Fourier series equation gives us the familiar rep- Electromagnetic Compatibility (EMC) resentation of a time domain function as a collection of EMC is a well-established engineering specialty for fundamental and harmonic frequencies. Although the the measurement, troubleshooting and prevention of Fourier series is infinite, for many signals, sufficient accu- unwanted radiation from electronic equipment. Digital racy can be obtained by using a finite number of terms. At equipment is particularly important, because the pulse some point, the coefficients (An, Bn) are small enough to train of bits has significant energy at frequencies that are be ignored. However, there are various waveforms that several multiples of the clock rate, as illustrated in the require a large number of terms before the coefficient val- previous discussion. ues reach small enough values. In the past, when digital equipment was just begin- One such waveform is a repeating pattern of very ning to become commonplace, clock rates were in the tens short pulses (Figure 1a), where each pulse has a duration of MHz. EMC concerns were limited to frequencies below of t and the repetition rate is t. In this case, the coeffi- 100 MHz. Remedies for excessive radiation were relative- cients will occur every 1/tin frequency, with amplitudes ly easy, using bypass capacitors, shielding, ferrite absorb- that fit into an “envelope” defined by a (sinx)/x function. ing materials and good layout techniques. The first zero crossing will be at a frequency of 1/t, desig- Now, with clock rates into the GHz region, significant nated f1 in Figure 1b. energy is present well into the microwave frequency If the pulse is very short (t << t), the number of discrete range. At higher frequencies, the energy is more difficult frequency terms between zero and f1 will be very large. to contain. Subtle factors like slots in an enclosure, length Several zero crossing periods are required before the of individual component interconnections and radiation amplitude of the coefficients is small enough to ignore. directly from (supposedly) terminated p.c. board traces all This illustrates that a large number of Fourier series are potential causes for unwanted radiation. terms may be required to accurately represent a relatively simple waveform. Digital Signal Integrity Without exploring the mathematics any further, it These same high frequency factors affect the accuracy should be apparent that digital signals require analysis of the actual pulse train of digital signals as they are con- in the frequency domain as well as the more familiar time ducted from component-to-component, or within compo- domain. We’ll review a few areas where this analysis is of nents themselves. These frequency-domain issues that current interest. affect digital signals have not been familiar to digital

56 High Frequency Electronics High Frequency Design TIME & FREQUENCY engineers until relatively recently. Some of the major fac- Time Domain Analysis of RF Signals tors are: The greatest need for information may be in the high- speed digital realm, but there are significant areas where • Reflected waves from circuit board traces that are not time domain techniques are essential to understanding designed as transmission lines and/or not properly ter- current RF/microwave communications. minated. One of these areas is reducing distortion to meet emis- • Reflected waves from discontinuities—bends in p.c. sions mask standards. Feedback, feedforward and predis- board traces, thickness changes due to components and tortion techniques all require accurate timing, and may solder, integrated circuit pads and internal connec- include adaptive capabilities that require signal sam- tions, and gaps in the ground plane layer. pling, analog-to-digital conversion, computation and digi- • Reduced signal voltage amplitude due to dielectric tal-to-analog conversion. Accuracy and timing in such losses, radiation losses and impedance mismatches. systems is critical. • Unequal propagation delays in parallel busses. Another interesting area requiring both time and fre- • Timing errors. quency analysis is propagation analysis. The time delays • Crosstalk between signal lines and other types of self- and amplitude variations in multipath signals, particu- interference. larly in a mobile environment, create a number of errors and distortions that require a combination of time and These issues have been widely discussed among the high- frequency techniques to identify and correct. Engineers level digital engineers involved in high speed computing, working in this area will be familiar with histograms, communications and digital imaging. However, the infor- waterfall displays and other methods that capture fre- mation is less well-known than it should be to many engi- quency data over time. neers. In some cases, the old saying that digital engineers Finally, new techniques such as polar modulation sep- don’t understand RF has been turned around—RF engi- arate a signal into phase and amplitude components that neers incorporating high-speed digital components in are primarily frequency domain and time domain, respec- their wireless products may not understand the full range tively. Extracting those signal components and recon- of digital issues. structing them in a high-efficiency power amplifier com- bines many of the issues discussed in this article.

58 High Frequency Electronics