22-Elec-A3 Signals and Communications · May 2015

Question 7 of 7: Speech Scrambler — Spectral Inversion and its Inverse

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 7: Speech Scrambler — Spectral Inversion and its Inverse (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A baseband message \(m(t)\) band-limited to \(B\) Hz; the required mapping is spectral inversion within the band, \(f \mapsto B - f\), with the output again baseband of bandwidth \(B\).

Find. A linear (time-varying) system realising the inversion, and the system that undoes it.

Approach. Recognise the mapping as a single-sideband frequency translation: mixing with a carrier at \(B\) places a reversed copy of the band in \(0\) to \(B\), which a low-pass filter then isolates.

1. Choose the carrier. Multiplying by \(2\cos(2\pi B t)\) produces the sum and difference bands, $$m(t)\cdot 2\cos(2\pi Bt) \;\longleftrightarrow\; M(f-B) + M(f+B)$$ A message component at frequency \(f\) therefore appears at \(B + f\) (upper sideband) and at \(|B - f|\) (lower sideband). For \(0 \lt f \lt B\) the lower sideband lands at \(B - f\), which is exactly the required mapping, and as \(f\) sweeps from \(0\) to \(B\) the image sweeps from \(B\) down to \(0\) — the band is reversed end for end.
2. Remove the unwanted sideband. The upper sideband occupies \(B\) to \(2B\) and must be discarded, so a low-pass filter with cut-off \(B\) follows the multiplier. The scrambler is thus a balanced modulator (a DSB-SC mixer) with carrier frequency exactly equal to the signal bandwidth, followed by a low-pass filter: $$\boxed{\,m(t) \to \times\,2\cos(2\pi B t) \to \text{LPF}(B) \to m_s(t)\,}$$ The factor 2 in the local carrier simply restores the amplitude that the product-to-sum identity halves.
   Figure Q7.1 — the scrambler: balanced modulator at carrier frequency B followed by a low-pass filter of cut-off B. The descrambler is identical.
   Figure Q7.2 — the mapping f → B - f reverses the baseband spectrum end for end while preserving its width.
3. Confirm the mapping. Take a single tone \(m(t) = \cos(2\pi f_1 t)\) with \(0 \lt f_1 \lt B\). Then $$2\cos(2\pi f_1t)\cos(2\pi Bt) = \cos\big(2\pi(B-f_1)t\big) + \cos\big(2\pi(B+f_1)t\big)$$ and the low-pass filter keeps only the first term, at \(B - f_1\), as required. Because the system is a multiplication by a fixed waveform followed by a fixed filter, it is linear (though time-varying), so this tone-by-tone result extends to any message by superposition.
4. Descrambler (part b). The mapping \(f \mapsto B - f\) is its own inverse: applying it twice returns \(B - (B-f) = f\). Therefore the descrambler is an identical system — multiply the scrambled signal by \(2\cos(2\pi Bt)\) and low-pass filter at \(B\): $$\boxed{\,m_s(t) \to \times\,2\cos(2\pi B t) \to \text{LPF}(B) \to m(t)\,}$$ This self-inverse property is the practical charm of the analogue speech inverter: the same circuit board serves at both ends of the link. The only requirement is that the two oscillators run at the same frequency \(B\); a frequency error \(\delta\) shifts every recovered component by \(\delta\) and, because the shift is additive rather than multiplicative, destroys the harmonic relationships that make speech sound natural — which is precisely why an uncorrelated listener hears the scrambled version as unintelligible in the first place.
5. Practical note on security. Frequency inversion is a very weak cipher: the transformation depends on no key beyond the publicly known bandwidth \(B\), so anyone with a mixer can undo it. It is intended to defeat casual eavesdropping on an analogue radio channel, not a determined attacker, and modern systems use digital encryption of the sampled speech instead.