Amplitude and angle modulation, receivers and multiplexing

Carrier plus two sidebands, each of amplitude $\mu A_c/2$. Bandwidth $B=2f_m$ (for a general message, $B=2W$ where $W$ is the highest message frequency).

These power relations hold for any $\mu$. Without overmodulation ($\mu \le 1$) the efficiency is at most $1/3$, reached at $\mu =1$. Each sideband carries ${\mu }^2{P}_c/4$. In terms of antenna (load) current, $I_t=I_c\sqrt{1+{\mu }^2/2}$.

For a message made of tones of different frequencies, each with its own index ${\mu }_i$. Use ${\mu }_t$ in the power and efficiency formulas. The condition ${\mu }_1+{\mu }_2+\cdots \le 1$ is enough to keep the envelope from going negative.

Use when $m(t)$ is not a single tone. Assumes $m(t)$ has zero mean; $\overline{m^2}$ is the mean-square value (power) of the message.

$A_{\max }$ and $A_{\min }$ are the maximum and minimum of the envelope, for a single tone with $\mu \le 1$. For $\mu >1$ (overmodulation) the envelope is distorted and an envelope detector fails.

The first condition (with $f_m$ replaced by $W$ for a general message) lets the capacitor smooth out the carrier ripple yet follow the envelope. The second is the condition, for a single tone of index $\mu <1$, that avoids diagonal clipping (the capacitor discharging slower than the envelope falls).

After low-pass filtering, a square-law detector's output is proportional to the squared envelope. Besides the message it contains a term in $m^2(t)$; for a single tone this is a second harmonic of $f_m$ at $\mu /4$ of the fundamental, so a small $\mu$ keeps the distortion small.

Power in a $1\Omega$ load, for a zero-mean message. The spectrum has no carrier component. The envelope is $A_c|m(t)|$, not $m(t)$, so an envelope detector cannot be used; DSB-SC needs coherent detection.

DSB-SC input, local carrier of unit amplitude $\cos ({\omega }_{c}t+\varphi )$ or $\cos ({\omega }_c+\Delta \omega )t$, followed by a low-pass filter. The output vanishes at $\varphi =90^{\circ }$ (quadrature null effect), and a frequency error gives a beating (fading) output. For SSB, a phase error causes phase distortion of the output, and a frequency error shifts every output tone by $\Delta f$. A Costas loop or a squaring loop recovers a synchronous carrier for DSB-SC.

Upper sign gives USB, lower sign gives LSB. $\hat{m}(t)$ is the Hilbert transform of $m(t)$ and has the same power as $m(t)$. Power in a $1\Omega$ load. Coherent detection with a unit-amplitude local carrier gives $\frac{A_c}{4}m(t)$.

$k_p$ in rad/V, $k_f$ in Hz/V. Instantaneous frequency: FM $f_i=f_c+k_{f}m(t)$; PM $f_i=f_c+\frac{k_p}{2\pi }\frac{dm}{dt}$. FM by $m(t)$ is PM by $\int mdt$, and PM by $m(t)$ is FM by $dm/dt$.

In FM, $\Delta f$ depends only on the amplitude of the message and $\beta$ falls as $f_m$ rises. In PM, $\beta$ depends only on the amplitude and $\Delta f$ grows with $f_m$.

Spectral lines at $f_c\pm n{f}_m$ with amplitudes $A_c|J_n(\beta )|$. The carrier amplitude $A_c{J}_0(\beta )$ vanishes at $\beta \approx 2.405,5.520,8.654,\dots$

In a $1\Omega$ load. Total power is independent of $\beta$; modulation only redistributes power from the carrier to the sidebands. Power in the carrier plus $k$ pairs of sidebands is $\frac{A_c^2}{2}[J_0^2+2{\sum }_{n=1}^k{J}_n^2]$.

An approximation; for a single tone it contains roughly 98 percent of the total power. For $\beta \ll 1$, $B\approx 2f_m$ (narrowband); for $\beta \gg 1$, $B\approx 2\Delta f$ (wideband). For a general message use $B=2(\Delta f+W)=2W(D+1)$, with peak deviation $\Delta f=k_f\max |m|$ and deviation ratio $D=\Delta f/W$.

Carrier plus two sidebands of amplitude $A_c\beta /2$, like AM, but the sideband sum is in phase quadrature with the carrier (the lower sideband's sign is reversed relative to AM), so the envelope is nearly constant.

Coherent detection for DSB-SC and SSB; envelope detection for AM above threshold, with $\mu \le 1$, so its figure of merit is at most $1/3$; FM above threshold without pre-emphasis. For a general zero-mean message of power $P$: AM gives $k_a^{2}P/(1+k_a^{2}P)$ and FM gives $3k_f^{2}P/W^2$. FM beats fully modulated ($\mu =1$) AM when $\frac{3}{2}{\beta }^2>\frac{1}{3}$, that is $\beta >\sqrt{2}/3\approx 0.47$. Above threshold, FM trades bandwidth for noise performance.

The noise at an FM discriminator's output has a PSD proportional to $f^2$, so high-frequency message components suffer most. The transmitter boosts the high frequencies (pre-emphasis) and the receiver cuts them by the same amount (de-emphasis). The overall response to the message is flat, while the high-frequency noise is reduced. A $75\mu \text{s}$ time constant corresponds to $f_1\approx 2.12$ kHz.

Ratio of the output noise power without de-emphasis to that with it, for the $f^2$ noise PSD over $|f|\le W$, assuming pre-emphasis does not change the transmitted peak deviation. Example: $W=15$ kHz and $f_1=2.1$ kHz give $I\approx 21.3$ (about 13.3 dB).

$f_s$ is the wanted station frequency. The image station lies at $f_{LO}+f_{IF}$ and also mixes down to $f_{IF}$, causing interference. With a low-side LO ($f_{LO}=f_s-f_{IF}$), $f_{im}=f_s-2f_{IF}$.

For a single tuned circuit of loaded quality factor $Q$ at the RF stage. $\alpha$ is a voltage ratio; in dB it is $20{\log }_{10}\alpha$. With identical tuned stages before the mixer, their ratios multiply. A larger IF moves the image farther from $f_s$ and improves image rejection; a smaller IF makes adjacent-channel selectivity easier to achieve.

High-side LO. For LC tuning $f\propto 1/\sqrt{C}$, so the capacitance ratio is the square of the frequency ratio. With the LO above the signal the required ratio is much smaller: for 540-1650 kHz and a 455 kHz IF it is about 2.1 high-side against about 14 low-side, which is why the high-side LO is standard in AM broadcast receivers.

$B_{guard}$ is the guard band between adjacent channels. For $N$ identical SSB channels of bandwidth $W$, $B_{total}=NW+(N-1)B_{guard}$. For DSB-SC or AM channels each $B_k=2W$.

$N$ channels, each band-limited to $W$ and sampled at $f_{\text{samp}}$ (the frame rate), at or above the Nyquist rate $2W$. The minimum (Nyquist) transmission bandwidth is half the pulse rate: $N{f}_{\text{samp}}/2$ for PAM-TDM and $R_b/2$ for binary PCM-TDM. Add synchronisation bits per frame if given.