The Hilbert transform is a linear operation applied to real signals. In practical terms, the Hilbert transform translates into a phase shift of -90º at the positive frequencies (and +90º at the negative frequencies) that make up the signal. The relevance of the Hilbert transform in telecommunication engineering is due to its contribution in obtaining spectrally efficient signals. For example in the generation of signals with single sideband spectrum or in the generation of the analytical signal.
Next, the basic mathematics of the Hilbert transform and its generic effects on signals in the time and frequency domains are reviewed. Finally, its main applications are discussed. The text is organized with the following table of contents:
1. The Mathematics of Hilbert Transform
In this section it is shown that the Hilbert transform \footnotesize \hat{s}(t) of a real signal s(t) is an operator \footnotesize \mathcal H() equivalent to a linear, noncausal, time invariant filter [1] [2]. The mathematics of the Hilbert transform are explained in both the time and frequency domains. For this purpose, in a generic way, the following notation is used:
\begin{equation} s(t) \xtofrom[\mathcal{H}^{-1}]{\mathcal{H}} \hat{s}(t) \end{equation}
\begin{equation} \hat{s}(t) = s(t) * h_{\mathcal{H}}(t) \end{equation}
\begin{equation} \hat{S}(f) = S(f) · H_{\mathcal{H}}(f) \end{equation}
Where \footnotesize h_{\mathcal{H}}(t) represents the impulse response of the Hilbert transform operator in the time domain, and \footnotesize H_{\mathcal{H}}(f) represents its transfer function in the frequency domain.
1.1 Frequency Domain
The analysis in the frequency domain allows a more intuitive interpretation of the Hilbert transform. Given a real signal whose (hermitic) spectrum is S(f), the transformed spectrum \footnotesize \hat{S}(f) essentially experiences a phase shift of 90°. Specifically, and given the hermiticity of a real signal, this is a phase shift of -90º at positive frequencies and +90º at negative frequencies. Mathematically, therefore, the transfer function of the Hilbert transform is:
\begin{equation} H_{\mathcal{H}}(f) = -j·sgn(f) = \begin{cases} -j = e^{-j \frac{\pi}{2}} &\text{for } f > 0 \\ 0 &\text{for } f = 0 \\ +j= e^{+j \frac{\pi}{2}} &\text{for } f < 0 \end{cases} \end{equation}
Thus, the Hilbert transform of the basic trigonometric functions is as follows:
\begin{equation} \cos(2\pi ft) \xtofrom[\mathcal{H}^{-1}]{\mathcal{H}} \sin(2\pi ft) \xtofrom[\mathcal{H}^{-1}]{\mathcal{H}} -\cos(2\pi ft) \xtofrom[\mathcal{H}^{-1}]{\mathcal{H}} - \sin(2\pi ft) \xtofrom[\mathcal{H}^{-1}]{\mathcal{H}} \cos(2\pi ft) \end{equation}
Note that, in practical terms, the above equation implies that if a signal is represented as a sum of positive frequency components, its Hilbert transform is obtained by adding a phase shift of -90º to these components.
1.2 Time Domain
The impulsional response of the Hilbert transform is:
\begin{equation} h_{\mathcal{H}}(t) = \cfrac{1}{\pi t} \end{equation}
And, consequently, it is satisfied that its Fourier transform is equal to the transfer function mentioned in the previous section:
\begin{equation} \cfrac{1}{\pi t} \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} -j·sgn(f) \end{equation}
Proving the above equation directly involves employing the concept of the Cauchy Principal Value, which is beyond the scope of this text. Instead, an indirect demonstration is made based on the Duality Property of the Fourier transform, so that it must be satisfied that:
\begin{equation} -j·sgn(t) \xtofrom[\mathcal{F}^{-1}]{\mathcal{F}} -\cfrac{1}{\pi f}\end{equation}
To prove the above equation, the following concepts must be taken into account:
- The derivative of the sign function is related to the Dirac delta δ(t) according to the following equation:
\begin{equation} \frac{\partial}{\partial t}sgn(t) = 2\delta(t) \end{equation}
- The Fourier transform of the Dirac delta function is equal to unity so that, from equation (9), it follows:
\begin{equation} {\mathcal{F}} \left[\frac{\partial}{\partial t}sgn(t)\right] = 2 \end{equation}
- The Fourier transform of the derivative of a generic function g(t) is related to the transform of the function itself such that:
\begin{equation} {\mathcal{F}} \left[\frac{\partial}{\partial t}g(t)\right]= j2\pi f{\mathcal{F}}[g(t)] \end{equation}
Applying the generic equation (11) to the function sgn(t) and comparing the result with equation (10), equation (8) follows directly as it was intended to demonstrate.
2. Hilbert Transform Effects on Signals
This section shows that the effects of the Hilbert transform depend on the spectrum of the original signal. In particular, the cases of baseband and bandpass signals are distinguished. In practice, Hilbert transform applications focus on the spectrum and the spectral efficiency, so the effects on the signal in the time domain may not be relevant.
For a more generic description of the effect of constant frequency phase shift on real signals, please consult this link.
2.1 Baseband Signals
The following image shows the spectrum of a real baseband signal before and after applying the Hilbert transform. The phase shifts of -90º at positive frequencies and 90º at negative frequencies are observed.
Since each of the frequencies that make up the signal suffers a constant phase shift, which is not linear with frequency, the appearance of the resulting time signal is different from that of the original signal. In other words, phase distortion has occurred. The following graph shows an example of a real baseband signal and its Hilbert transform, in the time domain, where the change of aspect can be appreciated:
2.2 Bandpass Signals
In line with the previous example, the application of the Hilbert transform to a real bandpass signal produces the effect on the spectrum shown in the following image:
With a reasoning equivalent to the baseband example, it could be deduced that the resulting signal is different from the original signal, presenting phase distortion. The following graph illustrates this behavior:
However, in the case of bandpass signals there is an important clarification. As shown in the image, the envelope of the transformed signal (in black) is equal to the envelope of the original signal. The envelope is the modulating signal, or the signal that is typically intended to be communicated. The effect of the Hilbert transform can be understood as equivalent to a 90º phase shift in the transmitter’s carrier. Therefore, once the receiver locked to the received signal, the demodulated envelope would be the same as the transmitted envelope. In other words, there would be no distortion in the communication.
3. Applications of the Hilbert Transform in Communications
In telecommunications engineering, the Hilbert transform is a fundamental tool for processing and obtaining spectrally efficient signals. [3]. This section briefly reviews their main applications.
3.1 Single Side Band Spectrum (SSB)
In this case, the Hilbert transform is used to reduce the transmission bandwidth of a bandpass signal.
3.1.1 Schematic Representation
The following graph is used to illustrate this strategy:
The starting point is a real baseband signal s(t). For simplicity, and without loss of generality, the baseband signal in the image consists of a single tone. As seen in the upper branch, when the baseband signal modulates a carrier, frequency components are obtained on both sides of the center frequency ωc (red and green components in the image). This is known as a double sideband signal (DSB).
However, when adding or subtracting the double-sideband signals obtained by quadrature modulating the original signal s(t) and its Hilbert transform \footnotesize \hat{s}(t) , a single sideband spectrum (SSB) is obtained. Indeed, because the sum of two components with an offset of π radians cancels out, a spectrum can be obtained that only includes frequencies below or above the center frequency.
3.1.2 Mathematical Representation
It is evident that the above reasoning can be extended to a generic real baseband signal s(t) with a given bandwidth. In this way the Hilbert transform allows to obtain an SSB bandpass signal by the scheme shown in the image. Mathematically:
\begin{equation} s_{SSB}(t) = s(t)\cos(\omega_ct) \pm \hat{s}(t)\sin(\omega_ct) \end{equation}
The above scheme can be implemented both in an analog form (typically using 90º hybrid couplers and IQ mixers) and with digital signal processing.
3.2 Quadrature Signals
The Hilbert transform is also used in the generation of quadrature signals, i.e. in the complex plane. The advantages of generating and processing these signals are briefly explained below.
3.2.1 Analytic Signal
Given a real bandpass signal s(t), its analytical signal sa(t) is complex and incorporates only the positive frequencies of s(t). In addition sa*(t), which is also complex, incorporates only the negative frequencies of s(t).
3.2.1.1 Mathematical Representation
The analytic signal with the properties described above is obtained by means of the following equation:
\begin{equation} s_{a}(t) = s(t)+j\hat{s}(t) \end{equation}
\begin{equation} s_{a}^*(t) = s(t)-j\hat{s}(t) \end{equation}
It is practically immediate to demonstrate that negative frequencies have been eliminated in the generation of the signal sa(t). Mathematically, applying (3) and (4) in (13) gives that:
\begin{equation} S_a(f) = \begin{cases}S(f)+j[-jS(f)] =2S(f)&\text{for } f >0 \\ S(f)+j0 =S(f)&\text{for } f =0 \\ S(f)+j[jS(f)]=0 &\text{for } f<0 \end{cases} \end{equation}
Similarly:
\begin{equation} S_a^*(f) = \begin{cases}0&\text{for } f >0 \\ S(f)&\text{for } f =0 \\ 2S(f) &\text{for } f<0 \end{cases} \end{equation}
3.2.1.2 Spectral Representation
Below is an example representing the spectrum of a real bandpass signal and its analytic signal:
The main advantage of the analytic signal is to eliminate the negative frequencies of a real signal, which can be considered superfluous due to Hermitic symmetry. Switching to complex notation facilitates many mathematical manipulations, especially in modulation and demodulation techniques. After processing the corresponding application, taking the real part of the post-processed analytical signal allows to obtain the real result signal with all its frequencies, positive and negative.
3.2.2 Complex Envelope
The complex envelope is obtained from the analytic signal. It represents the baseband signal resulting from moving the analytic signal from its center frequency to DC.
3.2.2.1 Mathematical Representation
To transfer a bandpass signal from a center frequency, without creating additional replicas, it is necessary to multiply it by a frequency phasor. Assuming that the analytic signal is centered at the carrier frequency ωc, the complex envelope can be obtained by the following operation:
\begin{equation} s_{a\downarrow}(t) = s_{a}(t)e^{-j\omega_c t} \end{equation}
\begin{equation} s_{a\uparrow}(t) = s_{a}^*(t)e^{j\omega_c t} \end{equation}
3.2.2.2 Spectral Representation
Below is an example representing the spectrum of a real bandpass signal and its complex envelope:
A very important advantage over the previous analytic signal is obtained: the bandwidth of the signal can be significantly reduced. Therefore signal processing can be performed at a lower sampling rate. However, recovering the bandpass signal is more complex because it also requires a frequency shift:
\begin{equation} s(t) = \real[s_{a}(t)] = \real[s_{a\downarrow}(t)e^{j\omega_c t}] \end{equation}
\begin{equation} s(t) = \real[s_{a}^*(t)] = \real[s_{a\uparrow}(t)e^{-j\omega_c t}] \end{equation}
More details on the analytic signal and the complex envelope are provided in this link. And, in particular, the implementation of the complex envelope in real-time applications is detailed in this link.
4. Conclusions
The conclusions of the text are as follows:
- The Hilbert transform is a linear operator that produces a phase shift of -90º in the (positive) frequencies of a signal.
- The effect of the Hilbert transform in the time domain depends on the signal spectrum. While in baseband cases the appearance of the signal changes completely, in bandpass cases the signal envelope remains unchanged.
- The main application of the Hilbert transform in communications is the generation of spectrally efficient signals: single sideband signal, analytical signal and complex envelope.
Bibliography
[1] Communication Systems, A. Bruce Carlson.
[2] Signals and Systems, A. V. Openheim.
[3] Hilbert Transform in Signal Processing, Stephan Hahn.
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