3 Time- vs. Frequency-Domain
Electrical signals - periodic or non-periodic - can be measured in the time domain (e.g. with an oscilloscope) or in the frequency domain (e.g. with a spectrum analyzer). This means that an electrical signal can be described either in the time-domain or in the frequency-domain. The time-domain representation helps you to determine the signal integrity (ringing, reflections), whereas the frequency domain representation helps you to determine at which frequencies a signal may lead to radiated emissions.
Fourier analysis is an integral transform and 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 (notation: x(t)○─●X(ω)). 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.
The picture below shows a representation of a square wave signal (1V amplitude) with the sum of only 4 harmonic sine waves and a direct current (DC) component of 0.5V. For the representation of an ideal square wave, an indefinite number of sine waves would be necessary (because the rise- and fall-time of an ideal square wave is 0sec).
The different variants of integral transforms are presented in the list below. All variants have in common that they present the necessary math for converting a signal from the time- to the frequency-domain and vice versa: X(ω)●─○x(t). Every integral transform variant has its own field of application.
Fourier series. Time signal = continuous, periodic. Frequency spectrum = discrete. Formulas for the calculation of the complex Fourier coefficients cn of a signal x(t) are presented below [3.1]:
T is the period length of the signal x(t) in [sec], f0 is the first harmonic of the signal x(t) in [Hz], ω0 is the first harmonic of the signal x(t) in [rad], j=√(-1) is the complex number and n stands for the n-th harmonic frequency [0, 1, 2, ...].
Fourier transform. Time signal = continuous, non-periodic. Frequency spectrum = continuous. Formulas of the Fourier transform of a signal x(t) (left) and the inverse-Fourier transform (right) are given below [3.2]:
X(ω) is the Fourier transform (spectrum) of x(t), ω is the frequency in [rad], j=√(-1) is the complex number and t the time in [sec].
Discrete Fourier transform (DFT). Time signal = discrete, periodic. Frequency spectrum = discrete. The DFT is also applied to non-periodic signals in the time domain by periodically continuing the non-periodic signals in order to make them computable with the DFT. The DFT is by far the most common method of modern Fourier analysis. The Fast Fourier Transformation (FFT) is a fast algorithm for calculating the DFT (in case the bock length N is a power of two) [3.3].
X is the spectrum of x and X[k] is the k-th sample of the spectrum at ωk. x[n] is the n-th sample of the discrete time signal, which is sampled with the sampling time Ts [sec] and the sampling frequency fs [Hz].
Discrete-time Fourier transform (DTFT). Time signal = discrete, non-periodic. Frequency spectrum = continuous. The DTFT can be viewed as the form of the DFT when its length N approaches infinity [3.4].
X is the spectrum of x. x[n] is the n-th sample of the discrete time signal, which is sampled with the sampling time Ts [sec] and the sampling frequency fs [Hz]. ῶ denotes the continuous normalized radian frequency variable [-π...+π]. ῶ is the product of the radian frequency ω [rad] and the sampling time Ts.
Laplace transform. The Laplace transformation belongs - like the Fourier analyses - to the group of integral transformations. It is mentioned here for the sake of completeness. The Laplace transform is used for system analysis (e.g. control systems, filters), whereas the Fourier transform is used for signal analysis [3.5].
X is the Laplace transform of x. s=σ+jω is a complex variable with a real part σ and a imaginary part with the radian frequency ω [rad]. In case of σ=0, the Laplace transform reduces to the Fourier transform.
Z-transform. The Z-transform is the time-discrete counterpart to the Laplace transform with z = e^(sTsampling) = e^((σ+jω)Tsampling) [3.6].
X is the Z-transform of x. z is a complex variable. x[n] is the n-th sample of the discrete time signal, which is sampled with the sampling time Ts [sec] and the sampling frequency fs [Hz].
Spectra of digital waveforms.
Waveforms of primary importance in electronic circuits are clock and data signals. Digital signals in the time domain can be represented by trapezoid-shaped pulses with a period time T [sec], a pulse width tpw [sec], a rise-time tr [sec] and a fall-time tf [sec]. The two pictures below show an extract of a digital waveform in the time-domain and an amplitude frequency spectrum of a digital signal with its envelope curve [3.7].
The frequency spectrum envelope of a trapezoid-waveform signal compared to an ideal square wave signal is shown below. It can be seen, that there is no -40dB drop-off for the ideal square wave. Instead, the frequency spectrum envelope of an ideal square wave drops constantly with -20dB/decade.
The picture below compares the frequency spectrum of two trapezoid waveforms with different pulse-width tpw [sec], but with identical periodicity T [sec], rise- and fall-time tr [sec] and tf [sec] and amplitude A [V]. It can be seen how the duty-cycle D=tpw/T [%] of a digital signal influences the frequency spectrum. However, 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]). The clock signal with a 90% duty-cycle has more power, but this power adds to the DC component (0 Hz).
The picture below compares the frequency spectrum of two trapezoid waveforms with different rise- and fall-time tr [sec] and tf [sec], but with identical periodicity T [sec], pulse-width tpw [sec] and amplitude A [V]. It can be seen that a reduction of the rise- and fall-time leads to lower amplitudes at the higher frequencies in the spectrum (and therefore to potentially lower radiated emissions). In this example, an increase of the rise- and fall-time [sec] by factor 10, reduces the amplitude of the high-frequency harmonics ( f > 32MHz) also by factor 10 (20 dB).
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 [W] in the frequency component is bigger than 50% of the power in an ideal square wave's signal with the same amplitude A [V] and duty-cycle D=tpw/T [%]. A drop in 50% of the power [W] is the same as a drop of 70% in amplitude [V] or a drop of 3dB.
The rule of thumb for calculating the bandwidth [Hz] - or the highest significant sine wave frequency - of a trapezoid digital signal is [3.8]:
Where t10%-90% is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of a digital signal in [sec].
The picture below shows the frequency spectrum envelop curves of an ideal square wave (t10%-90%=0) and a real waveform (trapezoid with t10%-90%≠0). The bandwidth of the real waveform can be found at f=0.35/t10%-90%. Note: It is assumed that there is no ringing in the real waveform. In case of ringing, the frequency spectrum envelope for f>1/(πt10%-90%) would not drop off with -40dB/decade.