Keywords

5.1 Fourier Analysis

Figure 5.1 presents measurement instruments for measuring electrical signals in the time-domain (oscilloscope) and the frequency-domain (spectrum analyzer). Fourier analysis refers to the mathematical principle that every signal can be represented by the sum of simple trigonometric functions (sine, cosine, etc.). The Fourier analysis enables a transformation of a signal in the time-domain x(t) to a signal in the frequency-domain X(ω), where ω = 2πf. In other words, a Fourier analysis is a mathematical operation for calculating the frequency-domain representation (frequency spectrum) of a signal in the time-domain. Two common Fourier analysis notations are:

$$\displaystyle \begin{aligned} \mathscr{F}\{x(t)\}=X(\omega) \end{aligned}$$

where:

  • x(t) = a signal in the time-domain

    Fig. 5.1
    figure 1

    Time-domain and frequency-domain measurement equipment. However, some oscilloscopes do also have built-in Fourier analysis functionality (FFT). (a)Time domain: high-end oscilloscope R&S®RTO64 by Rohde & Schwarz. (b) Frequency domain: signal and spectrum analyzer R&S®FSVA3030 by Rohde & Schwarz

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  • X(ω) = the Fourier transform of x(t) (frequency-domain)

Figure 5.2 shows a representation of a square wave signal (1 V amplitude) with the sum of only four harmonic sine waves (first, third, fifth, and seventh) and a direct current (DC) component of 0.5 V. For representing an ideal square wave, an indefinite number of sine waves would be necessary (because the rise- and fall-time of an ideal square wave are 0 s).

Fig. 5.2
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Fourier analysis of a square wave; up to the seventh harmonic

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There are different types of integral transforms. These transforms have in common that they define the necessary math for converting a signal from time- to the frequency-domain and vice versa. Every integral transform has its field of application:

  • Fourier Series . Continuous and periodic signals.

  • Fourier Transform . Continuous, nonperiodic signals.

  • Discrete Fourier Transform (DFT) . Discrete and periodic signals.

  • Fast Fourier Transform (FFT) . The FFT is a DFT. The FFT is an implementation of the DFT for fast and efficient computation.

  • Discrete-time Fourier Transform (DTFT). Discrete, nonperiodic signals.

  • Laplace Transform . Control systems and filter design.

  • Z-Transform . Time-discrete control systems and filter design.

Figure 5.3 shows some common integral transforms used in the field of EMC. Further details about the different integral transforms can be found in the Appendix F.

Fig. 5.3
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Time-domain to frequency-domain transforms

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5.2 Frequency Spectra of Digital Signals

Digital signals like clocks and data interfaces are the primary cause for EMC emissions of electronic circuits and electrical equipment and systems. Digital signals in the time-domain can be represented by trapezoid-shaped pulses with a period time T [sec], a pulse width t pw [sec], a rise-time t r [sec], and a fall-time t f [sec]. Figure 5.4 shows an extract of a digital waveform and the corresponding frequency spectrum (Fourier analysis). It can be seen that the pulse width t pw [sec] defines the frequency at which the amplitude spectrum starts to drop with −20 dB/decade, whereas the minimum fall- and/or rise-time defines the frequency at which the spectrum starts to drop with −40 dB/decade. Figure 5.5a compares the frequency spectrum of clock signals with different duty-cycles [%] (pulse widths). It is remarkable that a clock signal with a 90 % duty-cycle has a lower amplitude of the first harmonic than a clock signal with a 50 % duty-cycle (of the same frequency [Hz] and with the same rise-/fall-time [sec]). However, as a 90 % clock signal has more power, this power adds to the DC component (0 Hz). For EMC design engineers, Fig. 5.5b is even more important than Fig. 5.5a because it shows how high-frequency harmonics of a clock signal can be reduced effectively. In this example, an increase of the rise- and fall-time [sec] by factor 10 reduces the amplitude of the high-frequency harmonics (f > 32 MHz) also by factor 10 (20 dB).

Fig. 5.4
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Digital signal in the time- and frequency-domain. (a) Digital waveform with period time T [sec], pulse width t pw [sec], rise-time t r [sec] and fall-time t f [sec]. (b) Amplitude frequency spectrum of a digital signal and its envelope curve [3]

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Fig. 5.5
figure 5

Clock signal parameter analysis in the frequency-domain [1]. (a) Frequency spectrum of 100 kHz clock signals with rise-/fall-times of 10 nsec and duty-cycles of 50% vs. 90%. (b) Frequency spectrum of 100 kHz clock signals with 50% duty-cycle and rise-/fall-times of 10 nsec vs. 100 nsec

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5.3 Bandwidth of Digital Signals

Bandwidth [Hz] of a digital signal means what is the highest significant sine wave frequency component in the digital signal? Significant in this case means that the power in [W] in the frequency component is bigger than 50 % of the power in an ideal square wave’s signal with the same amplitude A in [V] and duty-cycle D = (t pw∕T) in [%]. A drop in 50 % of the power [W] is the same as a drop of 70 % in amplitude [V] or a drop of 3 dB.

The rule of thumb for calculating the bandwidth [Hz]—or the highest significant sine wave frequency—of a trapezoid digital signal is [2]:

$$\displaystyle \begin{aligned} B =\frac{0.35}{t_{10-90\%}} \end{aligned} $$
(5.1)

where:

  • B = bandwidth = highest significant sine wave frequency (harmonic) in a digital signal [Hz]

  • t 10−90% = rise- and/or fall-time (whichever is smaller) from 10 % to 90 % of the slope of a digital signal in [sec]

Figure 5.6 shows the frequency spectrum envelop curves of an ideal square wave \(t_r = t_f = {0}^{{\sec }}\) and a trapezoid with \(t_r > {0}^{{\sec }}\) and \(t_f > {0}^{{\sec }}\). The 3 dB-bandwidth of the trapezoid waveform can be found at f 3dB = 0.35∕t 10−90%.

Fig. 5.6
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The 3dB-bandwidth of a clock signal with a pulse width of t pw [sec], rise- and fall-times of t r = t f [sec]

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Note: it is assumed that there is no ringing in the waveform and t r = t f. In case of ringing, the frequency spectrum envelope for f > 1∕(πt r) would not drop off with −40 dB/decade. Instead, an increase in the frequency spectrum is assumed at the ringing frequency (see Fig. 5.7).

Fig. 5.7
figure 7

Simulation data of a signal with ringing (blue) vs. without ringing (gray). Both signals have an amplitude of 3.3 V, a fundamental frequency of 100 kHz, and a rise-/fall-time of 10 nsec [1]. (a) Time-domain. (b) Frequency-domain

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5.4 Ringing and Frequency Spectrum

Impedance mismatch along a transmission line could cause ringing . One could argue that ringing has nothing to do with EMC and that ringing is a signal integrity topic. However, this is not entirely correct. Due to ringing, higher frequency components (harmonics) are introduced, and it may be possible that the conductor—where the signal is propagating along—acts as an efficient antenna at these higher frequencies.

In the example of Fig. 5.7, the harmonics around 50 MHz show as much as 100 times higher voltage amplitudes (40 dB) in the signal with ringing than compared to the signal without ringing. This example illustrates that impedance matching does also matter for EMC and that EMC design engineers must know which signal connections must be considered as transmission lines and which not.

5.5 Summary

  • Time- vs. frequency-domain. Electrical signals have a time-domain x(t) and a frequency-domain X(ω) representation.

  • Measurement methods. Time-domain representations are typically measured with an oscilloscope and frequency-domain representations with a spectrum analyzer.

  • Bandwidth of digital signals.

    $$\displaystyle \begin{aligned} B \approx 0.35/t_{10-90\%} \end{aligned} $$
    (5.2)

    where:

    • B = highest significant sine wave frequency (harmonic) in a digital signal [Hz]

    • t 10−90% = rise- and/or fall-time (whichever is smaller) from 10 % to 90 % of the digital signal in [sec]