The Phasing Method and Hilbert Transforms for Single-Sideband Modulation

The first U.S. patent application for single-sideband (SSB) modulation was filed in 1915 by John Renshaw Carson. Carson proposed the filter method, which produces SSB signals by removing one sideband of a double-sideband (DSB) signal via filtering. Because the upper and lower sidebands of a DSB signal are separated by only a small amount, however, this method requires extremely sharp filters.

In 1924, Carson’s colleague Ralph V. L. Hartley introduced a second approach to SSB generation that did away with the requirement for a sharp bandpass filter. This approach, known as the phasing method, aims to cancel the unwanted sideband by modifying the circuit architecture rather than first generating both sidebands and then using a bandpass filter at the output to eliminate the undesired sideband.

In this article, we’ll delve into the phasing method. Along the way, we’ll also learn about the Hilbert transform. This useful signal processing tool, which we discussed briefly in the previous article, plays a key role in the phasing method by applying a broadband, 90 degree phase lag to each frequency component of the input signal.

 

DSB Modulation: A Review

To understand the phasing method, we first need to review the DSB modulation of a single-tone input. Assume that the message signal is given by:

$$m(t) ~=~ A_m \cos ( \omega_m t)$$

Equation 1.

 

The spectrum of the baseband signal consists of two impulses at ±f_m:

$$M(f)~=~\frac{A_m}{2} \Big [ \delta(f~-~f_m)~+~\delta(f~+~f_m) \Big ]$$

Equation 2.

 

With double-sideband suppressed-carrier (DSB-SC) modulation, the message signal is multiplied by a sinusoidal carrier wave given by:

$$c(t) ~=~ A_c \cos ( \omega_c t)$$

Equation 3.

 

Multiplying the message signal by the carrier wave produces:

$$\begin{eqnarray}s(t) ~=~ m(t) c(t) ~&=&~ A_m \cos( \omega_m t) ~\times~ A_c \cos ( \omega_c t) \\~&=&~ \frac{1}{2} A_m A_c \Big ( \cos \big (( \omega_c ~+~ \omega_m)t \big ) ~+~ \cos \big (( \omega_c ~-~ \omega_m)t \big ) \Big )\end{eqnarray}$$

Equation 4.

 

Figure 1 shows the spectrum of the DSB-SC signal for our single-tone input. The upper and lower sidebands of the signal are denoted by USB and LSB, respectively.

 

Figure 1. The DSB-SC signal’s output spectrum when the message signal is a cosine function at f_m.

 

Multiplying two sinusoidal waves at ⍵_m and ⍵_c produces new frequencies that are the sum (⍵_c + ⍵_m) and difference (⍵_c – ⍵_m) of the original frequencies.

 

The Basic Idea of the Phasing Method

The goal of SSB signal generation is to produce only the sum (USB) or difference (LSB) term from the inputs cos(⍵_mt) and cos(⍵_ct). Applying basic trigonometric identities, the sum and difference terms can be expanded as:

$$\cos \big ( (\omega_c ~\pm~ \omega_m)t \big )~=~ \cos( \omega_c t) \cos( \omega_m t ) ~\mp~ \sin( \omega_c t) \sin( \omega_m t)$$

Equation 5.

 

Using Equation 5 to produce just the sum or difference term requires us to generate sin(⍵_mt) from the message signal, m(t) = cos(⍵_mt).

A cosine wave is a sine wave shifted by 90 degrees. We say a cosine wave leads the sine wave by 90 degrees, whereas a sine wave lags a cosine wave by 90 degrees. In mathematical terms, we have:

$$\sin( \omega_m t) ~=~ \cos (\omega_m t ~-~ \frac{\pi}{2} ) \quad \text{and} \quad \sin( \omega_c t) ~=~ \cos (\omega_c t ~-~ \frac{\pi}{2} )$$

Equation 6.

 

This method therefore requires phase-shift networks that provide a precise phase shift of –90 degrees.

Figure 2 shows the block diagram of an SSB modulator built around Equation 5.

 

Figure 2. Block diagram of an SSB modulator that produces a lower-sideband signal.

 

This circuit generates only the lower sideband. If we replace the adder with a subtractor, it will generate the upper sideband instead.

Although the above result was obtained for a special case of m(t) = cos(ω_mt), it is valid for an arbitrary message waveform. This is because using a Fourier series allows us to represent any practical signal as a sum of sine and cosine functions.

 

The Hilbert Transform

The Hilbert transform is named after David Hilbert, a distinguished German mathematician whose work has had a profound and lasting impact. Among his other contributions, Hilbert developed the theory of Hilbert spaces and laid the groundwork for functional analysis.

The Hilbert transform shifts all positive frequency components by –90 degrees and all negative frequency components by +90 degrees. The spectral amplitudes are not affected. In other words, the Hilbert transform corresponds to a linear filter with the following frequency response:

$$H(f) ~=~ \begin{cases} \begin{array}{rc}j && f ~<~ 0 \\0 && f ~=~ 0 \\-j && f ~>~ 0 \end{array}\end{cases}$$

Equation 7.

 

Figure 3 shows the magnitude and phase response of the Hilbert transform.

 

Figure 3. The magnitude (a) and phase (b) of the Hilbert transform’s frequency response.

 

In the time domain, the Hilbert transform can be associated with a linear filter with the following impulse response:

$$h(t) ~=~ \frac{1}{\pi t}$$

Equation 8.

 

The Hilbert transform of a cosine function is a sine wave, as we’ll discuss below.

 

Finding the Hilbert Transform of Cosine and Sine Functions

Because the Hilbert transform alters the phase of positive and negative frequency components differently, it may not be immediately apparent that the Hilbert transform of a cosine function results in a sine wave. To resolve any ambiguity, let’s find the Hilbert transform of the cosine and sine functions.

A real-valued cosine function with frequency ⍵_m can be written as:

$$\cos( \omega_m t) ~=~ \frac{1}{2} \big ( e^{j \omega_m t} ~+~ e^{ -j \omega_m t} \big )$$

Equation 9.

 

The first term on the right-hand side of the equation represents a positive frequency. It undergoes a phase shift of –90 degrees. The second term corresponds to a negative frequency and is shifted in phase by +90 degrees. The Hilbert transforms of cos(⍵_mt) and sin(⍵_mt) can therefore be calculated as follows:

$$\begin{eqnarray}HT \big \{ \cos( \omega_m t) \big \} ~&=&~ \frac{1}{2} \big (-j e^{j \omega_m t} ~+~ je^{ -j \omega_m t} \big ) \\~&=&~ \frac{-j}{2} \big ( e^{j \omega_m t} ~-~ e^{ -j \omega_m t} \big ) \\ ~&=&~ \frac{1}{2j} \big ( e^{j \omega_m t} ~-~ e^{ -j \omega_m t} \big ) \\~&=&~ \sin ( \omega_m t)\end{eqnarray}$$

Equation 10.

 

and:

$$\begin{eqnarray}HT \big \{ \sin( \omega_m t) \big \} ~&=&~ HT \big \{\frac{1}{2j} \big ( e^{j \omega_m t} ~-~ e^{ -j \omega_m t} \big ) \big \} \\~&=&~ \frac{1}{2j} \big ( -je^{j \omega_m t} ~-~ je^{ -j \omega_m t} \big ) \\ ~&=& ~\frac{-1}{2} \big ( e^{j \omega_m t} ~+~ e^{ -j \omega_m t} \big ) \\~&=&~ -\cos ( \omega_m t)\end{eqnarray}$$

Equation 11.

 

As expected, the Hilbert transform converts cos(⍵_mt) to sin(⍵_mt). It also converts sin(⍵_mt) to –cos(⍵_mt). From this, we see that the Hilbert transform applies the required 90 degree phase lag to each frequency component of the input signal.

As a side note, Equations 10 and 11 also reveal a general property of the Hilbert transform. If m_h(t) is the Hilbert transform of m(t), then the Hilbert transform of m_h(t) is –m(t).

 

Applying a Hilbert Transform to SSB Modulation

Let’s return to Figure 2 for a moment. We now have the mathematical background to better describe this system. By incorporating a Hilbert transform, we can obtain a functional block diagram of the phasing method (Figure 4).

 

Figure 4. The functional block diagram of the SSB phasing method.

 

Above, the Hilbert transform takes the place of the first –90 degree phase shifter. Its input and output are in quadrature, meaning that they’re 90 degrees out of phase with respect to each other.

The circuit employs two multipliers. The carrier wave and the message signal are applied to one path. On the other path, the carrier and message signals are shifted in phase by 90 degrees before they’re applied. When the outputs of the two paths are combined, the phase-shifting action cancels out one of the sidebands.

In practice, the phase-shift networks are implemented through a passive network or digital signal processing techniques. The multipliers are commonly built around balanced modulators. The extent of carrier suppression is influenced by the accuracy of the balanced modulators; the suppression level of the unwanted sideband is determined by the phase-shifting block.

From Figure 4, we can describe SSB signals in the time domain as follows:

$$s(t) ~=~ m(t) \cos(\omega_c t) ~\mp~ m_h(t) \sin(\omega_c t)$$

Equation 12.

 

where m_h(t) is the Hilbert transform of the message signal, m(t). The plus sign produces the lower sideband; the minus sign produces the upper sideband. Note that we’re assuming A_c = 1 for the sake of simplicity.

 

Wrapping Up

The phasing method produces a low-level SSB signal. Linear RF amplifiers then enhance the SSB output’s power before it’s sent to the transmitting antenna. In the next article, we’ll explore the phasing method further with the help of some graphical representations.

 

This article is Part 11 of a series on amplitude modulation in RF systems. A complete list of articles in this series is provided below:

1. Introduction to Modulation Techniques in RF Systems
2. Understanding Double-Sideband Suppressed-Carrier Modulation
3. Understanding Conventional Amplitude Modulation
4. Understanding the Square-Law Modulator for Generating AM Signals
5. Introduction to the Balanced Modulator for AM Signals
6. How Do Switching Modulators Generate AM Signals?
7. Understanding How Ring Modulators Produce AM Signals
8. Four Interesting AM Modulation Circuits You Should Know About
9. Demodulating Double-Sideband AM Signals
10. Introduction to Single-Sideband Modulation: The Filter Method
11. The Phasing Method and Hilbert Transforms for Single-Sideband Modulation
12. A Visual Approach to Understanding the Phasing Method for SSB Modulation
13. How Phasors Help Us Understand Bandpass Signals
14. Introduction to Weaver’s Method for SSB Signal Generation
15. Exploring the Operation of the Weaver Modulator for Single-Sideband Modulation

 

All images used courtesy of Steve Arar