Noise in Communication Systems

The threshold effect in FM is the sharp drop in output signal-to-noise ratio of an FM receiver when the input carrier-to-noise ratio (CNR) falls below a certain value, called the threshold, typically about 10 dB (about 13 dB in some texts).

Above threshold

When the carrier is much stronger than the noise, noise only causes small phase perturbations. The output SNR grows linearly with input CNR and FM gives a large improvement over AM:

$$(SNR)_o=\frac{3}{2}\beta^2\,(SNR)_c\quad(\text{tone modulation})$$

Below threshold

 (SNR)o dB
   |              /  FM above threshold
   |             /
   |            /
   |     ------/  <- knee (threshold, ~10 dB CNR)
   |    /
   |   /   sharp fall below threshold
   +---------------------------> (SNR)c dB

When noise amplitude occasionally exceeds the carrier amplitude, the resultant phasor can encircle the origin, causing a sudden $2\pi$ phase jump. The discriminator output (derivative of phase) then shows sharp impulses, heard as clicks. As CNR falls, clicks become frequent, the noise rises rapidly, and the signal is "captured" by noise. The output SNR then falls much faster than the input CNR and the FM advantage is lost.

Consequences and remedies

• Large $\beta$ improves SNR above threshold but needs more bandwidth, so more noise enters and the threshold rises; this sets a practical limit on $\beta$.
• Threshold extension (lowering the threshold by a few dB) uses an FM demodulator with feedback (FMFB) or a phase-locked loop demodulator, which track the signal with a narrower effective bandwidth.
• Pre-emphasis/de-emphasis improves SNR above threshold but does not remove the threshold.

A matched filter is the linear filter that maximises the peak output signal-to-noise ratio at the sampling instant when a known pulse is received in additive white Gaussian noise (AWGN). It is used as the optimum detector in digital receivers.

Derivation

Consider a known pulse $g(t)$, $0\le t\le T$, plus white Gaussian noise $w(t)$ of power spectral density $N_0/2$, applied to a linear time-invariant filter with impulse response $h(t)$ and transfer function $H(f)$.

 x(t) = g(t) + w(t) --> [ h(t) ] --> y(t) = go(t) + n(t)
                                         |
                                     sample at t = T

Signal output at the sampling instant:

$$g_o(T)=\int_{-\infty}^{\infty}H(f)G(f)e^{j2\pi fT}df$$

Noise output power (white noise through $H(f)$):

$$E[n^2(t)]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df$$

Peak pulse SNR to be maximised:

$$\eta=\frac{\lvert g_o(T)\rvert^2}{E[n^2(t)]}=\frac{\left|\int H(f)G(f)e^{j2\pi fT}df\right|^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df}$$

Schwarz inequality: for any $\phi_1(f),\phi_2(f)$,

$$\left|\int\phi_1\phi_2\,df\right|^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$$

with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=G(f)e^{j2\pi fT}$:

$$\left|\int H(f)G(f)e^{j2\pi fT}df\right|^2\le\int\lvert H(f)\rvert^2df\int\lvert G(f)\rvert^2df$$

Substituting into $\eta$:

$$\eta\le\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert G(f)\rvert^2df=\frac{2E}{N_0}$$

using Rayleigh's energy theorem $E=\int\lvert G(f)\rvert^2df=\int g^2(t)dt$.

Optimum filter: the maximum $\eta_{max}=2E/N_0$ is reached when equality holds:

$$H_{opt}(f)=k\,G^*(f)\,e^{-j2\pi fT}$$

Impulse response: take the inverse Fourier transform:

$$\begin{aligned} h_{opt}(t)&=k\int_{-\infty}^{\infty}G^*(f)e^{-j2\pi f(T-t)}df\\ &=k\left[\int_{-\infty}^{\infty}G(f)e^{j2\pi f(T-t)}df\right]^*\quad(\text{real }g)\\ &=k\,g(T-t) \end{aligned}$$

So the impulse response of the optimum filter is the time-reversed and delayed (by $T$) version of the input signal, scaled by $k$. Because it is "matched" to the signal, it is called the matched filter.

  g(t)                     h(t) = g(T - t)
   |\                          /|
   | \                        / |
   |  \                      /  |
   +---+-----> t        +---+---+----> t
   0   T                0       T
  (flip about t = 0, then shift right by T)

Properties

• Maximum output SNR $=2E/N_0$ depends only on the signal energy, not on its shape.
• $\lvert H_{opt}(f)\rvert=k\lvert G(f)\rvert$: the filter passes strongly where the signal spectrum is strong.
• The output $g_o(t)$ is proportional to the autocorrelation of $g(t)$ shifted by $T$, peaking at $t=T$; hence a matched filter is equivalent to a correlator (multiply by $g(t)$ and integrate over $0$ to $T$).
• The delay $T$ makes $h(t)$ causal (zero for $t<0$).

Example

For a rectangular pulse $g(t)=A$, $0\le t\le T$, the matched filter is $h(t)=kA$ for $0\le t\le T$ (an integrate-and-dump filter). The output is a triangle peaking at $t=T$ with value $kA^2T=kE$, and $\eta_{max}=2A^2T/N_0$.

Optimum detector

An optimum detector is a receiver that makes decisions with the minimum probability of error for given signals and noise. In AWGN this is achieved by first passing the received signal through a filter that maximises the output SNR at the sampling instant (the matched filter or an equivalent correlator), then comparing the sample with a threshold (maximum-likelihood decision).

Matched filter impulse response

Consider a known pulse $g(t)$, $0\le t\le T$, plus white Gaussian noise $w(t)$ of power spectral density $N_0/2$, applied to a linear time-invariant filter with impulse response $h(t)$ and transfer function $H(f)$.

 x(t) = g(t) + w(t) --> [ h(t) ] --> y(t) = go(t) + n(t)
                                         |
                                     sample at t = T

Signal output at the sampling instant:

$$g_o(T)=\int_{-\infty}^{\infty}H(f)G(f)e^{j2\pi fT}df$$

Noise output power (white noise through $H(f)$):

$$E[n^2(t)]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df$$

Peak pulse SNR to be maximised:

$$\eta=\frac{\lvert g_o(T)\rvert^2}{E[n^2(t)]}=\frac{\left|\int H(f)G(f)e^{j2\pi fT}df\right|^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df}$$

Schwarz inequality: for any $\phi_1(f),\phi_2(f)$,

$$\left|\int\phi_1\phi_2\,df\right|^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$$

with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=G(f)e^{j2\pi fT}$:

$$\left|\int H(f)G(f)e^{j2\pi fT}df\right|^2\le\int\lvert H(f)\rvert^2df\int\lvert G(f)\rvert^2df$$

Substituting into $\eta$:

$$\eta\le\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert G(f)\rvert^2df=\frac{2E}{N_0}$$

using Rayleigh's energy theorem $E=\int\lvert G(f)\rvert^2df=\int g^2(t)dt$.

Optimum filter: the maximum $\eta_{max}=2E/N_0$ is reached when equality holds:

$$H_{opt}(f)=k\,G^*(f)\,e^{-j2\pi fT}$$

Impulse response: take the inverse Fourier transform:

$$\begin{aligned} h_{opt}(t)&=k\int_{-\infty}^{\infty}G^*(f)e^{-j2\pi f(T-t)}df\\ &=k\left[\int_{-\infty}^{\infty}G(f)e^{j2\pi f(T-t)}df\right]^*\quad(\text{real }g)\\ &=k\,g(T-t) \end{aligned}$$

So the impulse response of the optimum filter is the time-reversed and delayed (by $T$) version of the input signal, scaled by $k$. Because it is "matched" to the signal, it is called the matched filter.

  g(t)                     h(t) = g(T - t)
   |\                          /|
   | \                        / |
   |  \                      /  |
   +---+-----> t        +---+---+----> t
   0   T                0       T
  (flip about t = 0, then shift right by T)

Properties

• Maximum output SNR $=2E/N_0$ depends only on the signal energy, not on its shape.
• $\lvert H_{opt}(f)\rvert=k\lvert G(f)\rvert$: the filter passes strongly where the signal spectrum is strong.
• The output $g_o(t)$ is proportional to the autocorrelation of $g(t)$ shifted by $T$, peaking at $t=T$; hence a matched filter is equivalent to a correlator (multiply by $g(t)$ and integrate over $0$ to $T$).
• The delay $T$ makes $h(t)$ causal (zero for $t<0$).

Properties of the autocorrelation function

The autocorrelation (AC) function measures how similar a signal is to a time-shifted copy of itself. For a power signal or a wide-sense stationary (WSS) random process $X(t)$:

$$R_X(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}x(t)\,x(t+\tau)\,dt \quad\text{or}\quad R_X(\tau)=E[X(t)X(t+\tau)]$$

For an energy signal, $R_g(\tau)=\int_{-\infty}^{\infty}g(t)g(t+\tau)\,dt$.

Three (and more) important properties:

1. Even symmetry: $R_X(-\tau)=R_X(\tau)$.
2. Value at origin = mean-square value (power): $R_X(0)=E[X^2(t)]=P$ (for an energy signal, $R_g(0)=E$).
3. Maximum at origin: $\lvert R_X(\tau)\rvert\le R_X(0)$ for all $\tau$.
4. Fourier pair with PSD (Wiener–Khinchin): $S_X(f)=\int_{-\infty}^{\infty}R_X(\tau)e^{-j2\pi f\tau}d\tau$ and $R_X(\tau)=\int_{-\infty}^{\infty}S_X(f)e^{j2\pi f\tau}df$.
5. Periodicity: if $x(t)$ is periodic with period $T_0$, $R_X(\tau)$ is also periodic with $T_0$.
6. DC component: if $X(t)$ has mean $m$ and no periodic part, $R_X(\tau)\to m^2$ as $\tau\to\infty$.

Autocorrelation function of white noise

White noise is an idealised random noise whose power spectral density is flat (constant) over all frequencies, like white light containing all colours. Thermal and shot noise are nearly white up to very high frequencies ($\sim 10^{12}$ Hz), so the model is accurate for any practical receiver bandwidth. It is usually also assumed Gaussian, zero-mean (AWGN).

PSD (two-sided):

$$S_W(f)=\frac{N_0}{2}\quad\text{for all } f$$

where $N_0=kT_e$ (W/Hz), $k$ = Boltzmann constant and $T_e$ = equivalent noise temperature.

Autocorrelation (inverse Fourier transform of PSD, using $\mathcal{F}^{-1}\{1\}=\delta(\tau)$):

$$R_W(\tau)=\int_{-\infty}^{\infty}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\,\delta(\tau)$$

   S_W(f)                      R_W(tau)
     |                             ^  (N0/2) delta
 N0/2+-----------------            |
     |                             |
 ----+-----------------> f   ------+------> tau
     0                             0

Meaning: $R_W(\tau)=0$ for every $\tau\ne0$, so any two samples of white noise, however close, are uncorrelated (and independent if Gaussian). The total power $R_W(0)=\int S_W(f)df$ is infinite, which is why white noise is only a model; after any filter with finite bandwidth the power becomes finite.

The autocorrelation (AC) function measures how similar a signal is to a time-shifted copy of itself. For a power signal or a wide-sense stationary (WSS) random process $X(t)$:

$$R_X(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}x(t)\,x(t+\tau)\,dt \quad\text{or}\quad R_X(\tau)=E[X(t)X(t+\tau)]$$

For an energy signal, $R_g(\tau)=\int_{-\infty}^{\infty}g(t)g(t+\tau)\,dt$.

It tells how fast a signal changes: a slowly varying signal stays correlated for large $\tau$, a rapidly varying one loses correlation quickly. It is used to find the PSD, to detect periodic signals buried in noise, and in correlation receivers and radar.

Properties:

1. Even symmetry: $R_X(-\tau)=R_X(\tau)$.
2. Value at origin = mean-square value (power): $R_X(0)=E[X^2(t)]=P$ (for an energy signal, $R_g(0)=E$).
3. Maximum at origin: $\lvert R_X(\tau)\rvert\le R_X(0)$ for all $\tau$.
4. Fourier pair with PSD (Wiener–Khinchin): $S_X(f)=\int_{-\infty}^{\infty}R_X(\tau)e^{-j2\pi f\tau}d\tau$ and $R_X(\tau)=\int_{-\infty}^{\infty}S_X(f)e^{j2\pi f\tau}df$.
5. Periodicity: if $x(t)$ is periodic with period $T_0$, $R_X(\tau)$ is also periodic with $T_0$.
6. DC component: if $X(t)$ has mean $m$ and no periodic part, $R_X(\tau)\to m^2$ as $\tau\to\infty$.

Example: white noise has $R(\tau)=\frac{N_0}{2}\delta(\tau)$; samples at different times are uncorrelated.

Energy and power spectral density

Energy spectral density (ESD): for an energy signal $g(t)\leftrightarrow G(f)$, $\Psi_g(f)=\lvert G(f)\rvert^2$ (J/Hz). It shows how the energy is spread over frequency:

$$E=\int_{-\infty}^{\infty}\lvert g(t)\rvert^2dt=\int_{-\infty}^{\infty}\Psi_g(f)\,df$$

and $\Psi_g(f)$ is the Fourier transform of the energy autocorrelation $R_g(\tau)$.

Power spectral density (PSD): for a power signal, take a truncated version $g_T(t)$ (length $T$) with transform $G_T(f)$:

$$S_g(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}\ \ \text{(W/Hz)},\qquad P=\int_{-\infty}^{\infty}S_g(f)\,df$$

For a WSS random process, $S_X(f)=\lim_{T\to\infty}E[\lvert X_T(f)\rvert^2]/T$. Both ESD and PSD are real, even and non-negative.

Wiener–Khinchin relation (PSD $\leftrightarrow$ AC): for a power signal, $R_g(\tau)=\lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}g_T(t)g_T(t+\tau)dt$. The time-domain integral is $g_T(\tau)*g_T(-\tau)$ scaled by $1/T$, and $g_T(-t)\leftrightarrow G_T^*(f)$, so

$$\begin{aligned} \mathcal{F}\{R_g(\tau)\} &= \lim_{T\to\infty}\frac{1}{T}G_T(f)G_T^*(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}=S_g(f) \end{aligned}$$

Hence

$$S_g(f)=\int_{-\infty}^{\infty}R_g(\tau)e^{-j2\pi f\tau}d\tau,\qquad R_g(\tau)=\int_{-\infty}^{\infty}S_g(f)e^{j2\pi f\tau}df$$

and at $\tau=0$: $R_g(0)=\int S_g(f)df=P$.

Autocorrelation of white noise from its PSD

White noise is an idealised random noise whose power spectral density is flat (constant) over all frequencies, like white light containing all colours. Thermal and shot noise are nearly white up to very high frequencies ($\sim 10^{12}$ Hz), so the model is accurate for any practical receiver bandwidth. It is usually also assumed Gaussian, zero-mean (AWGN).

PSD (two-sided):

$$S_W(f)=\frac{N_0}{2}\quad\text{for all } f$$

where $N_0=kT_e$ (W/Hz), $k$ = Boltzmann constant and $T_e$ = equivalent noise temperature.

Autocorrelation (inverse Fourier transform of PSD, using $\mathcal{F}^{-1}\{1\}=\delta(\tau)$):

$$R_W(\tau)=\int_{-\infty}^{\infty}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\,\delta(\tau)$$

   S_W(f)                      R_W(tau)
     |                             ^  (N0/2) delta
 N0/2+-----------------            |
     |                             |
 ----+-----------------> f   ------+------> tau
     0                             0

Meaning: $R_W(\tau)=0$ for every $\tau\ne0$, so any two samples of white noise, however close, are uncorrelated (and independent if Gaussian). The total power $R_W(0)=\int S_W(f)df$ is infinite, which is why white noise is only a model; after any filter with finite bandwidth the power becomes finite.

Band-limited white noise (more practical): if $S(f)=\frac{N_0}{2}$ for $\lvert f\rvert\le B$ and zero elsewhere,

$$R(\tau)=\int_{-B}^{B}\frac{N_0}{2}e^{j2\pi f\tau}df=N_0B\,\frac{\sin2\pi B\tau}{2\pi B\tau}$$

Its power is finite, $R(0)=N_0B$, and samples spaced $1/2B$ apart are uncorrelated.

Error probability in a binary communication system

Model. In a binary system, symbol 1 is sent as $s_1(t)$ and 0 as $s_2(t)$, each lasting $T_b$, with equal probability. Channel adds white Gaussian noise $w(t)$ of PSD $N_0/2$. The receiver filter output is sampled at $t=T_b$:

 s_i+w  +--------+ y(t)  sample  y  +---------+
 ------>| filter |------>-/ ------->| y > lam?|--> 1/0
        +--------+       t=Tb       +---------+

The sample is $y=a_i+n$, where $a_1, a_2$ are the noise-free outputs ($a_1>a_2$) and $n$ is Gaussian with zero mean and variance $\sigma^2$. Conditional pdfs:

$$f(y\mid1)=\frac{1}{\sqrt{2\pi}\sigma}e^{-(y-a_1)^2/2\sigma^2},\qquad f(y\mid0)=\frac{1}{\sqrt{2\pi}\sigma}e^{-(y-a_2)^2/2\sigma^2}$$

   f(y|0)            f(y|1)
     .--.             .--.
    /    \     |     /    \
   /      \    |    /      \
  /     ...\...|.../...     \
 --------a2---lam---a1----------> y
        Pe0 = area right of lam under f(y|0)
        Pe1 = area left  of lam under f(y|1)

Decision rule: choose 1 if $y>\lambda$, else 0. For equal priors the optimum threshold is midway, $\lambda=\frac{a_1+a_2}{2}$.

Error when 0 is sent ($y>\lambda$): with $u=(y-a_2)/\sqrt2\sigma$,

$$\begin{aligned} P_{e0} &= \int_{\lambda}^{\infty}\frac{1}{\sqrt{2\pi}\sigma}e^{-(y-a_2)^2/2\sigma^2}dy=\frac{1}{\sqrt{\pi}}\int_{(\lambda-a_2)/\sqrt2\sigma}^{\infty}e^{-u^2}du \\ &= \frac12\operatorname{erfc}\left(\frac{a_1-a_2}{2\sqrt2\,\sigma}\right) \end{aligned}$$

By symmetry $P_{e1}$ (1 sent, $y<\lambda$) is the same. The average error probability is

$$P_e=\tfrac12P_{e0}+\tfrac12P_{e1}=\frac12\operatorname{erfc}\left(\frac{a_1-a_2}{2\sqrt2\,\sigma}\right)=Q\left(\frac{a_1-a_2}{2\sigma}\right)$$

where $\operatorname{erfc}(x)=\frac{2}{\sqrt\pi}\int_x^{\infty}e^{-u^2}du$ and $Q(x)=\frac12\operatorname{erfc}(x/\sqrt2)$.

With a matched filter (matched to $s_1-s_2$), the maximum of $\frac{(a_1-a_2)^2}{\sigma^2}$ is $\frac{2E_d}{N_0}$, where $E_d=\int_0^{T_b}[s_1(t)-s_2(t)]^2dt$. Therefore

$$P_e=\frac12\operatorname{erfc}\left(\sqrt{\frac{E_d}{4N_0}}\right)$$

$P_e$ depends only on the energy of the difference signal relative to $N_0$, not on pulse shape. Making the two signals as different as possible (antipodal) gives the lowest $P_e$.

($E_b$ = average energy per bit.)

Threshold effect in FM

Threshold effect in FM. The FM improvement $\text{SNR}_o\propto\beta^2$ holds only when the carrier is much stronger than the noise at the discriminator input. When the carrier-to-noise ratio (CNR) falls below a threshold, about 10 dB (typically 10–13 dB), the noise phasor sometimes becomes larger than the carrier and the resultant phasor encircles the origin, causing a sudden $2\pi$ phase jump.

  high CNR: noise wobbles     low CNR: resultant can swing
  the tip slightly            around origin -> 2pi jump
        .-.                         .---.
  0 -->( * )                   0 --( *   )
        '-'                         '---'

Each $2\pi$ jump produces an impulse (spike) in the discriminator output, heard as clicks. The click rate rises rapidly as CNR drops, so the output SNR falls much faster than linearly and the signal is mutilated. Above threshold, FM is excellent; below it, FM is worse than AM. Wider deviation (larger $\beta$) means larger $B_T$, more noise and a higher threshold.

 SNRo(dB)            . FM above threshold
     |             .   (slope 1, offset 3/2 beta^2)
     |           .
     |         .
     |       . <- threshold knee (CNR ~ 10 dB)
     |     .:
     |   . :     AM / baseband
     |  .  :
     +------------------> SNRi (dB)

Mitigation of threshold effect

Techniques to reduce (extend) the threshold:

1. FM feedback (FMFB) demodulator: the VCO output tracks the signal, so the IF bandwidth needed is reduced (compressed deviation); less noise enters, threshold lowered by about 5–7 dB.
2. Phase-locked loop (PLL) demodulator: the loop's narrow bandwidth tracks the instantaneous frequency and rejects noise outside it; gives a few dB threshold extension.
3. Pre-emphasis and de-emphasis: boost high message frequencies before modulation and cut them after detection; reduces the high-frequency (parabolic) noise, improving SNR by about 10–13 dB and helping operation near threshold.
4. Reducing IF bandwidth / modulation index where possible, or increasing transmitted power (raises CNR above threshold).
5. Diversity reception and limiters that keep CNR above threshold under fading.

Optimum detector

An optimum detector is a receiver (filter followed by sampler and threshold device) that makes the decision with the minimum probability of error. For a known pulse in additive white Gaussian noise, this is achieved by the filter that maximises the output signal-to-noise ratio at the sampling instant; this optimum filter is the matched filter.

Impulse response of the optimum detector in additive white noise

Set-up. The received signal is $x(t)=s(t)+w(t)$, $0\le t\le T$, where $s(t)$ is a known pulse and $w(t)$ is white noise of two-sided PSD $N_0/2$. It passes through an LTI filter $h(t)\leftrightarrow H(f)$ and is sampled at $t=T$:

 s(t)+w(t)    +--------+   y(t)    sample     y(T)
 ----------->|  h(t)  |---------->--/ ---->  decision
              +--------+           at t=T

Output $y(t)=s_o(t)+n_o(t)$ with

$$s_o(T)=\int_{-\infty}^{\infty}H(f)S(f)e^{j2\pi fT}df,\qquad E[n_o^2]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df$$

We want $H(f)$ that maximises the output peak signal-to-noise ratio

$$\eta=\frac{\lvert s_o(T)\rvert^2}{E[n_o^2]}=\frac{\left\lvert\int H(f)S(f)e^{j2\pi fT}df\right\rvert^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df}$$

Schwarz inequality: $\left\lvert\int\phi_1\phi_2\,df\right\rvert^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$, with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=S(f)e^{j2\pi fT}$:

$$\eta\le\frac{\int\lvert H\rvert^2df\int\lvert S\rvert^2df}{\frac{N_0}{2}\int\lvert H\rvert^2df}=\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert S(f)\rvert^2df=\frac{2E}{N_0}$$

(using Rayleigh's theorem, $E$ = pulse energy). The maximum is reached when

$$H_{opt}(f)=k\,S^*(f)\,e^{-j2\pi fT}$$

Impulse response. Take the inverse transform. For real $s(t)$, $S^*(f)=S(-f)$, which is the transform of $s(-t)$; the factor $e^{-j2\pi fT}$ is a delay of $T$:

$$\begin{aligned} h_{opt}(t) &= k\int_{-\infty}^{\infty}S^*(f)e^{-j2\pi fT}e^{j2\pi ft}df \\ &= k\int_{-\infty}^{\infty}S(-f)e^{-j2\pi f(T-t)}df \\ &= k\,s(T-t) \end{aligned}$$

So the optimum (matched) filter impulse response is the input pulse reversed in time and delayed by $T$. Choosing the sampling instant $T$ equal to the pulse duration makes $h(t)$ causal.

   s(t)                      h(t) = s(T - t)
    |\                          /|
    | \                        / |
    |  \                      /  |
 ---+---+---> t          ---+---+---> t
    0   T                   0   T

Results: maximum SNR $=2E/N_0$ depends only on pulse energy, not on pulse shape; the filter's frequency response is the conjugate of the pulse spectrum, so it passes strongly the frequencies where the signal is strong.

Receiver model. The received signal $s(t)$ plus white noise ($N_0/2$) passes through a band-pass (IF) filter of bandwidth $B_T$, then a demodulator and a low-pass filter of bandwidth $W$ (message bandwidth).

 s(t)  +   +-------+    +-------+   +-----+
 ---->(+)-->|  BPF  |--->| demod |-->| LPF |--> y(t)
       ^    |  B_T  |    |       |   |  W  |
      w(t)  +-------+    +-------+   +-----+

The filtered noise is narrowband and is written in in-phase/quadrature form

$$n(t)=n_c(t)\cos\omega_ct-n_s(t)\sin\omega_ct$$

where $n_c$ and $n_s$ are low-pass, each with the same power as $n(t)$: $\overline{n_c^2}=\overline{n_s^2}=\overline{n^2}=N_0B_T$.

Definitions. Message power $P=\overline{m^2(t)}$.

• Input SNR $\text{SNR}_i=\dfrac{\text{received signal power}}{\text{noise power in } B_T}$
• Output SNR $\text{SNR}_o=\dfrac{\text{message power at output}}{\text{noise power at output}}$
• Detection gain $\gamma_d=\dfrac{\text{SNR}_o}{\text{SNR}_i}$; figure of merit $=\dfrac{\text{SNR}_o}{\text{SNR}_{ref}}$, where $\text{SNR}_{ref}$ is the channel SNR measured in the message band $W$ (baseband reference). Both show how much the demodulator improves or worsens SNR.

SSB-SC (coherent detection). $s(t)=\frac{A_c}{2}[m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct]$, $B_T=W$.

• Signal power $=\frac{A_c^2}{4}\cdot\frac{P+P}{2}=\frac{A_c^2P}{4}$ (since $\overline{\hat m^2}=P$); noise in $W$ $=N_0W$, so $\text{SNR}_i=\dfrac{A_c^2P}{4N_0W}$.
• Multiply by $\cos\omega_ct$ and low-pass filter:

$$y(t)=\frac{A_c}{4}m(t)+\frac12n_c(t)$$

• Signal power $\frac{A_c^2P}{16}$, noise power $\frac{N_0W}{4}$, so $\text{SNR}_o=\dfrac{A_c^2P}{4N_0W}$

$$\gamma_{SSB}=\frac{\text{SNR}_o}{\text{SNR}_i}=1$$

With baseband reference ($\text{SNR}_{ref}=\frac{A_c^2P}{4N_0W}$, same transmitted power), figure of merit $=1$.

DSB-SC (coherent detection). $s(t)=A_cm(t)\cos\omega_ct$, $B_T=2W$.

• Signal power $=\frac{A_c^2P}{2}$; noise in $2W$ $=2N_0W$, so $\text{SNR}_i=\dfrac{A_c^2P}{4N_0W}$.
• Multiply $x(t)=s(t)+n(t)$ by $\cos\omega_ct$ and low-pass filter:

$$y(t)=\frac{A_c}{2}m(t)+\frac12n_c(t)$$

• $\text{SNR}_o=\dfrac{A_c^2P/4}{2N_0W/4}=\dfrac{A_c^2P}{2N_0W}$

$$\gamma_{DSB\text{-}SC}=\frac{\text{SNR}_o}{\text{SNR}_i}=2$$

The quadrature noise $n_s$ is rejected by coherent detection, giving a 3 dB gain. With baseband reference $\text{SNR}_{ref}=\frac{A_c^2P}{2N_0W}$, figure of merit $=1$.

Comparison:

The DSB-SC detection gain is twice that of SSB because the two sidebands add coherently (voltage) while the noise in them adds in power. But SSB lets in only half the noise bandwidth, so for the same transmitted power and same noise PSD, both give the same output SNR (figure of merit 1). SSB is preferred because it needs half the bandwidth.

In ASK (on–off keying) a carrier is switched on for bit 1 and off for bit 0. Coherent detection uses a correlator with a locally generated carrier of the same frequency and phase.

ASK (on–off keying) signals over $0\le t\le T_b$:

$$s_1(t)=A\cos\omega_ct\ \ (\text{bit }1),\qquad s_2(t)=0\ \ (\text{bit }0)$$

Coherent receiver (correlator = matched filter):

 r(t) -->(x)--> integrate --> sample --> y > lam ? 1 : 0
          ^       0..Tb       t=Tb
          |
       cos(wc t)  (local carrier, in phase)

Correlator output at $t=T_b$, with $r(t)=s_i(t)+w(t)$:

$$y=\int_0^{T_b}r(t)\cos\omega_ct\,dt=a_i+n$$

• Bit 1: $a_1=\int_0^{T_b}A\cos^2\omega_ct\,dt=\frac{AT_b}{2}$ (for $f_c$ an integer multiple of $1/T_b$).
• Bit 0: $a_2=0$.
• Noise: $n=\int_0^{T_b}w(t)\cos\omega_ct\,dt$ is Gaussian, mean 0, variance

$$\sigma^2=\int_0^{T_b}\!\!\int_0^{T_b}\frac{N_0}{2}\delta(t-u)\cos\omega_ct\cos\omega_cu\,dt\,du=\frac{N_0}{2}\cdot\frac{T_b}{2}=\frac{N_0T_b}{4}$$

Threshold midway: $\lambda=\frac{AT_b}{4}$. Using the binary result $P_e=\frac12\operatorname{erfc}\left(\frac{a_1-a_2}{2\sqrt2\sigma}\right)$:

$$\begin{aligned} \frac{a_1-a_2}{2\sqrt2\,\sigma} &= \frac{AT_b/2}{2\sqrt2\sqrt{N_0T_b/4}}=\frac{A\sqrt{T_b}}{2\sqrt{2N_0}}=\sqrt{\frac{A^2T_b}{8N_0}} \end{aligned}$$

Energy of a "1" pulse is $E_1=\frac{A^2T_b}{2}$; the average energy per bit (half the bits are 0) is $E_b=\frac{E_1}{2}=\frac{A^2T_b}{4}$. Hence $\frac{A^2T_b}{8N_0}=\frac{E_b}{2N_0}$ and

$$P_e(\text{ASK})=\frac12\operatorname{erfc}\left(\sqrt{\frac{E_b}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{N_0}}\right)$$

(In terms of peak energy $E_1$: $P_e=\frac12\operatorname{erfc}\sqrt{E_1/4N_0}$.)

Remarks: coherent ASK needs 3 dB more average power than BPSK for the same $P_e$, and the threshold depends on received amplitude, so ASK is sensitive to fading. Example: for $E_b/N_0=10$ (10 dB), $P_e=Q(\sqrt{10})=Q(3.16)\approx7.9\times10^{-4}$.

A matched filter is a linear filter whose impulse response is matched to (a time-reversed, delayed copy of) a known input pulse $s(t)$, $h(t)=k\,s(T-t)$, so that it gives the maximum possible peak signal-to-noise ratio at the sampling instant $t=T$ when the pulse is received in additive white noise. It is the optimum detector for digital signals in AWGN; in practice it is built as a correlator.

Set-up. The received signal is $x(t)=s(t)+w(t)$, $0\le t\le T$, where $s(t)$ is a known pulse and $w(t)$ is white noise of two-sided PSD $N_0/2$. It passes through an LTI filter $h(t)\leftrightarrow H(f)$ and is sampled at $t=T$:

 s(t)+w(t)    +--------+   y(t)    sample     y(T)
 ----------->|  h(t)  |---------->--/ ---->  decision
              +--------+           at t=T

Output $y(t)=s_o(t)+n_o(t)$ with

$$s_o(T)=\int_{-\infty}^{\infty}H(f)S(f)e^{j2\pi fT}df,\qquad E[n_o^2]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df$$

We want $H(f)$ that maximises the output peak signal-to-noise ratio

$$\eta=\frac{\lvert s_o(T)\rvert^2}{E[n_o^2]}=\frac{\left\lvert\int H(f)S(f)e^{j2\pi fT}df\right\rvert^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df}$$

Schwarz inequality: $\left\lvert\int\phi_1\phi_2\,df\right\rvert^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$, with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=S(f)e^{j2\pi fT}$:

$$\eta\le\frac{\int\lvert H\rvert^2df\int\lvert S\rvert^2df}{\frac{N_0}{2}\int\lvert H\rvert^2df}=\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert S(f)\rvert^2df=\frac{2E}{N_0}$$

(using Rayleigh's theorem, $E$ = pulse energy). The maximum is reached when

$$H_{opt}(f)=k\,S^*(f)\,e^{-j2\pi fT}$$

Impulse response. Take the inverse transform. For real $s(t)$, $S^*(f)=S(-f)$, which is the transform of $s(-t)$; the factor $e^{-j2\pi fT}$ is a delay of $T$:

$$\begin{aligned} h_{opt}(t) &= k\int_{-\infty}^{\infty}S^*(f)e^{-j2\pi fT}e^{j2\pi ft}df \\ &= k\int_{-\infty}^{\infty}S(-f)e^{-j2\pi f(T-t)}df \\ &= k\,s(T-t) \end{aligned}$$

So the optimum (matched) filter impulse response is the input pulse reversed in time and delayed by $T$. Choosing the sampling instant $T$ equal to the pulse duration makes $h(t)$ causal.

   s(t)                      h(t) = s(T - t)
    |\                          /|
    | \                        / |
    |  \                      /  |
 ---+---+---> t          ---+---+---> t
    0   T                   0   T

Results: maximum SNR $=2E/N_0$ depends only on pulse energy, not on pulse shape; the filter's frequency response is the conjugate of the pulse spectrum, so it passes strongly the frequencies where the signal is strong.

Example: for a rectangular pulse $s(t)=A$, $0\le t\le T$, $h(t)=kA$ for $0\le t\le T$, which is the same rectangle; it behaves as an integrate-and-dump circuit. The output is a triangle with peak $kA^2T$ at $t=T$.