---
title: "SSB in terms of Hilbert transform; SSB generation"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:ssb-hilbert-derivation-generation"
source: https://www.bctengineer.com/q/semester_6/communication_systems/ssb-hilbert-derivation-generation
site: BCT Engineering
---

# SSB in terms of Hilbert transform; SSB generation

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2079 Chaitra (CS I), 6+4 marks; 2068 Jestha (old course), 6+4 marks; 2067 Shrawan (CS I), 6+4 marks (asked 3 times)
- Video: [Introduction to Single Side Band Suppressed Carrier with Frequency Spectrum | COM SYS | Lec-56 |](https://www.youtube.com/watch?v=LtHVbGxLXWg) — R K Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/ssb-hilbert-derivation-generation
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Derive the equation for Single Side-Band modulated signal in terms of Hilbert Transformation of modulating signal m(t). Briefly discuss any one method of generating SSB signal.

## Short answer

s_USB = (Ac/2)[m(t)cos ωct - m̂(t)sin ωct]; s_LSB = (Ac/2)[m(t)cos ωct + m̂(t)sin ωct]. The phase-shift method uses a wideband 90° phase shifter to obtain m̂(t); balanced modulators and summation cancel one sideband.

## Answer

A **single-sideband (SSB)** signal contains only one sideband (upper or lower) of a DSB-SC signal, so it needs only bandwidth $W$, half that of DSB.

#### Derivation of SSB in terms of Hilbert transform
**Tools:** the Hilbert transform $\hat m(t)$ of $m(t)$ has $\hat M(f)=-j\,\text{sgn}(f)M(f)$. The **pre-envelope** (analytic signal) is

$$
m_+(t)=m(t)+j\hat m(t)\ \leftrightarrow\ M(f)+\text{sgn}(f)M(f)=2M(f)u(f)
$$

So $m_+(t)$ contains **only the positive-frequency** part of $M(f)$, doubled. Similarly $m_-(t)=m(t)-j\hat m(t)\leftrightarrow 2M(f)u(-f)$ contains only negative frequencies.

**Upper sideband:** the USB spectrum is $M(f)$ shifted right by $f_c$ for its positive half and left by $f_c$ for its negative half:

$$
S_{USB}(f)=\frac{A_c}{4}\left[2M(f-f_c)u(f-f_c)+2M(f+f_c)u(-f-f_c)\right]
$$

Taking the inverse transform term by term:

$$
\begin{aligned}
s_{USB}(t)&=\frac{A_c}{4}\left[m_+(t)e^{j2\pi f_ct}+m_-(t)e^{-j2\pi f_ct}\right]\\
&=\frac{A_c}{4}\left[(m+j\hat m)e^{j\omega_ct}+(m-j\hat m)e^{-j\omega_ct}\right]\\
&=\frac{A_c}{4}\left[2m(t)\cos\omega_ct-2\hat m(t)\sin\omega_ct\right]\\
&=\frac{A_c}{2}\left[m(t)\cos\omega_ct-\hat m(t)\sin\omega_ct\right]
\end{aligned}
$$

**Lower sideband:** using $m_-$ at $+f_c$ and $m_+$ at $-f_c$, in the same way:

$$
s_{LSB}(t)=\frac{A_c}{2}\left[m(t)\cos\omega_ct+\hat m(t)\sin\omega_ct\right]
$$

Combined: $s_{SSB}(t)=\frac{A_c}{2}\left[m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct\right]$ (– for USB, + for LSB).

**Check with a tone:** $m=A_m\cos\omega_mt$, $\hat m=A_m\sin\omega_mt$. Then $s_{USB}=\frac{A_cA_m}{2}[\cos\omega_mt\cos\omega_ct-\sin\omega_mt\sin\omega_ct]=\frac{A_cA_m}{2}\cos(\omega_c+\omega_m)t$, a single line at $f_c+f_m$, as expected.

#### Generation: phase-shift (phase discrimination) method
This method implements the above equation directly.

```text
           +->[Mod 1]<-- cos wct ---+
           |   m cos wct            |
 m(t)------+                   Carrier osc
           |                        |
           +->[-90 deg]->[Mod 2]<-[-90 deg]
              m^(t)       m^ sin wct
                              |
     [Mod1] --(+)-- [Mod2]: sum (-/+) -> SSB
```

1. Balanced modulator 1 multiplies $m(t)$ by $\cos\omega_ct$, giving DSB-SC $m\cos\omega_ct$.
2. A wideband $-90°$ phase shifter (Hilbert transformer) produces $\hat m(t)$; a second $-90°$ shifter gives $\sin\omega_ct$ from the carrier. Balanced modulator 2 gives $\hat m\sin\omega_ct$.
3. Subtracting the outputs gives USB; adding gives LSB, because one sideband adds in phase and the other cancels.

It needs no sharp sideband filter, so it works at any carrier frequency and with message spectra extending to low frequencies; its difficulty is building an exact $90°$ shifter over the whole audio band.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the pre-envelope m+(t) = m(t) + j m^(t) contain in the frequency domain?
   A. Only the negative-frequency part of M(f), doubled
   B. Only the positive-frequency part of M(f), doubled
   C. Both halves of M(f) unchanged
   D. Only the DC component of M(f)
   Answer: B. Since the Hilbert transform gives -j sgn(f)M(f), the sum M(f)+sgn(f)M(f) equals 2M(f)u(f), which is positive frequencies only. (Key point: Pre-envelope)

2. Which expression gives the upper-sideband SSB signal?
   A. (Ac/2)[m(t)cos wct + m^(t) sin wct]
   B. (Ac/2)[m(t)cos wct - m^(t) sin wct]
   C. Ac m(t) cos wct
   D. (Ac/2)[m(t) sin wct - m^(t) cos wct]
   Answer: B. Combining m+ e^{jwct} and m- e^{-jwct} terms gives (Ac/2)[m cos wct - m^ sin wct] for USB; the plus sign gives LSB. (Key point: USB equation)

3. For m = Am cos wm t (so m^ = Am sin wm t), what does the USB equation reduce to?
   A. (AcAm/2) cos(wc - wm)t
   B. (AcAm/2) cos(wc + wm)t
   C. AcAm cos wm t cos wc t
   D. (AcAm/2)[cos(wc-wm)t + cos(wc+wm)t]
   Answer: B. cos wm t cos wc t - sin wm t sin wc t = cos(wc+wm)t, a single line at fc+fm. (Key point: Tone check)

4. In the phase-shift method, how are USB and LSB selected?
   A. By switching a sharp band-pass filter
   B. By subtracting the two modulator outputs for USB and adding for LSB
   C. By changing the carrier frequency
   D. By changing the message amplitude
   Answer: B. One sideband adds in phase and the other cancels, so subtracting gives USB and adding gives LSB. (Key point: Phase-shift generation)

## Related in Communication Systems

- [VSB generation; VSB for TV; SSB applications](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Avsb-generation-tv-ssb-applications) — Communication Systems, chapter 2
- [FM expression after change of modulating signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-new-waveform-500hz-2khz) — Communication Systems, chapter 3
- [USB-SSB expression for a random band limited message](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ausb-ssb-random-message) — Communication Systems, chapter 2
- [Angle modulated signal 12 sin(10⁶t + 5 sin10⁴t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-12sin-numerical) — Communication Systems, chapter 3
- [Angle modulated signal 20cos(6×10⁸t + 7sin1250t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-20cos-6e8t) — Communication Systems, chapter 3
- [Phase shift method of SSB generation: pros and cons](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-phase-shift-pros-cons) — Communication Systems, chapter 2, asked 2×

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