Test: Matched Filter - Electronics and Communication Engineering (ECE) MCQ

Matched Filter:

• Matched Filter is used to produce an output in such a way that it maximizes the ratio of the output peak power to the mean noise power in its frequency response.
• The frequency response of Magnitude and phase angle of the Matched Filter varies uniformly with frequency.
•  The impulse response of the matched filter depends upon the signal shape. It is the mirror image of the received signal about a time instant.
• The Characteristics of the matched filter are matched with the transmitted data.
• The matched filter reduces INTERSYMBOL INTERFERENCE by attenuating the beginning and end of each symbol period.
• Matched Filter measures the correlation between incoming received message and its impulse response.

Important Points:

• A time delay is necessary for the specification of the filter for reasons of physical realizability since there can be no output from the filter until the signal is applied.
• The frequency-response function of the matched filter is the conjugate of the spectrum of the received waveform except for the phase shift exp (- j2Πft).
• This phase shift varies uniformly with frequency. Its effect is to cause a constant time delay.

The shown bock diagram will be considered as the receiver in digital communication.

The impulse response of the filter is calculated by:

h(t) = s*(T_b - t)
Calculation:
In the given signal the duration is T. So, T_b = T
The shifted signal by T is given by:
s(t+T)

The time-reversed signal will be

Option 3 is correct.

A matched filter is used to increase the signal to noise ratio (S/N) at the receiver by correlating with the known input signal.

If h(t) = si(T – t), maximum signal to noise ratio will be there at the output receiver.
Signal strength (s) = Energy of s(t)

The output SNR is given by:

In a matched filter the filter response is shaped according to the input signal waveform.
The peak amplitude of the output of the filter is proportional to the input signal energy, hence the probability of error depends on the signal energy.

y(t) = s(t) × h(t)
y(t) = s(t) × s(T_b – t)

1. The matched filter impulse response h(t) for input x(t) is h(t) = x(T-t).
2. For matched Filter The output has a duration that is twice the duration of the input signal (due to convolution) and its peak value is the energy of the input signal
Application: Since p(t) is a uniform signal, T_s time long, the matched filter h(f) will also be T_s time long. So, the convolution of  p(t) and h(t) will be 2T_s time long signal.
Now, the energy of p(t) will be calculated as:

E = 1
So, the peak value of y(t) will be 1.
Note: It is the rule that is always followed by outputs of matched filters. The output has a duration that is twice the duration of the input signal (due to convolution) and its peak value is the energy of the input signal.

Convolution of a signal x(t) with unit impulse δ(t) is the signal itself. i.e. x(t) ⊕ δ(t) = x(t)
Fourier transform of auto-correlation function of a power signal x(t) is power spectral density Sx(f). i.e. 
And E(x² (t)) = R_X (0)
The variance of the signal x(t) is defined as:

Fourier transform of unit impulse is 1.

Let n(t) be the input white noise with zero mean and  N₀/2 power spectral density.
Mean of the white noise = E(n(t)) = 0
Power spectral density is:

And the auto-correlation function is:

Let y_n(t) is the output noise.

Mean of the output noise:

The variance of the output noise is:

A matched filter is used to increase the signal-to-noise ratio (S/N) at the receiver end by correlating the filter impulse response with the known input signal.
i.e. provided h(t) = S_i*(T - t), where Signal S_i(t) exists for 0 ≤ t ≤ T, the filter is said to be a matched filter.
i.e. h(t) = (S_iR(T - t) + jS_iImg (T - t))*
⇒ h(t) = (S_iR(T - t) – jS_iImg(T - t))
Where, S_iR = Real part of S_i(t)
And S_iImg = Imaginary part of S_i(t).
Calculation:
Given, S_i(t) as shown;

So the real part of the filter impuse response must be –ve of the shifted part shown below,

And the imaginary part must be,

- [SiImg (T - t)] will be,

So, Option (4) is Correct.

A matched filter is used to increase the signal to noise ratio (S/N) at the receiver by correlating with the known input signal.

Provided h(t), which is the impulse response of the matched filter , where signal s_i(t) exists for 0 ≤ t ≤ T, the signal to noise ratio will be maximum at the receiver end.
For a real input signal,  
So, if h(t) = s_i(T – t), the maximum signal to noise ratio will be there at the output receiver.
Given,
S_i(t) as shown:

So, S_i(-t)

And S_i(-t+T) will be;