An optimum filter is such a filter used for acquiring a best estimate of desired signal from noisy measurement. It is different from the classic filters. These filters are optimum because they are designed based on optimization theory to minimize the mean square error between a processed signal and a desired signal, or equivalently provides the best estimation of a desired signal from a measured noisy signal.
It is pervasive that when we measure a (desired) signal d(n), noise v(n) interferes with the signal so that a measured signal becomes a noisy signal
x(n) x(n)=d(n)+v(n)
It is also very common that a signal d(n) is distorted in its measurement (e.g., an electromagnetic signal distorts as it propagates over a radio channel). Assuming that the system causing distortion is characterized by an impulse response of h (n) s , the measurement of d(n) can be expressed by the sum of distorted signal s(n) and noise
v(n) x(n)=s(n)+v(n)= h (n)∗ s d(n)+v(n)
where s(n)= h (n)∗ s d(n).
If both d(n) and v(n) are assumed to be wide-sense stationary (WSS) random processes, then x(n) is also a WSS process. The signals that we discuss in this chapter will be WSS if they are not specially specified. If signal d(n) and measurement noise v(n) are assumed to be uncorrelated (this is true in many practical cases), then r (k) = r (k) = 0.
In this case, the noisy signal,
x(n)= h (n)∗ s d(n)+v(n),
the relation of r (k) x with r (k ) d and r (k) v (the autocorrelations of x(n), d(n) and v(n), respectively) as follows,
For the noisy signal of the form x(n)= d(n)+v(n), a special case of where h (n) s =δ (n) and no distortion happens to d(n) in its measurement, we have
Optimum filtering is to acquire the best linear estimate of a desired signal from a measurement. The main issues in optimal filtering contain
• filtering that deals with recovering a desired signal d(n) from a noisy signal (or measurement) x(n);
• prediction that is concerned with predicting a signal d(n+m) for m>0 from observation x(n);
• smoothing that is an a posteriori form of estimation, i.e., estimating d(n+m) for m
If a filter produces an output in such a way that it maximizes the ratio of output peak power to mean noise power in its frequency response, then that filter is called Matched filter.
Frequency Response Function of Matched Filter
The frequency response of the Matched filter will be proportional to the complex conjugate of the input signal’s spectrum. Mathematically, we can write the expression for frequency response function, H(f) of the Matched filter as −
H(f)=GaS∗(f)e−j2πft1 ………..1
Where,
Ga is the maximum gain of the Matched filter
S(f) is the Fourier transform of the input signal, s(t)
S∗(f) is the complex conjugate of S(f)
t1 is the time instant at which the signal observed to be maximum
In general, the value of Ga is considered as one. We will get the following equation by substituting Ga=1in Equation 1.
H(f)=S∗(f)e−j2πft1 ……..2
The frequency response function, H(f) of the Matched filter is having the magnitude of S∗(f)and phase angle of e−j2πft1, which varies uniformly with frequency.
Impulse Response of Matched Filter
In time domain, we will get the output, h(t) of Matched filter receiver by applying the inverse Fourier transform of the frequency response function, H(f).
h(t)=
….….3
Substitute, Equation 1 in Equation 3.
h(t)=
⇒h(t)=
………4
We know the following relation.
S∗(f)=S(−f) ……..5
Substitute, Equation 5 in Equation 4.
h(t)=
⇒h(t)=
⇒h(t)=Gas(t1−t)
In general, the value of Ga is considered as one. We will get the following equation by substituting Ga=1.
h(t)=s(t1−t)
The above equation proves that the impulse response of Matched filter is the mirror image of the received signal about a time instant t1. The following figures illustrate this concept.
The received signal, s(t) and the impulse response, h(t) of the matched filter corresponding to the signal, s(t) are shown in the above figures.
In a digital transmission, BER is the number of bits with errors divided by the total number of bits that have been transmitted, received or processed over a given time period. That is
𝐵𝐸𝑅 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑖𝑡𝑠 𝑤𝑖𝑡ℎ 𝑒𝑟𝑟𝑜𝑟 / 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑖𝑡𝑠 𝑠𝑒𝑛𝑡
Bit error rate is a key parameter that is used in assessing the systems performance that transmits digital data from one location to another. When data is transmitted over a data link, there is a possibility of errors being introduced into the system. As a result, it is necessary to assess the performance of the system, and BER provides an ideal way in which this can be achieved. BER assesses performance of a system including the transmitter, receiver and the medium between the two
BER for BPSK modulation
In a BPSK system the received signal can be written as
Where 
and 
. The real part of the above equation is 
where 
In BPSK constellation is 
and 
is defined as 
and sometimes it is called SNR per bit.
With this definition we have
So the bit error probability is
This equation can be simplified using Q-function as
Where the Q function is defined as
BER for QPSK modulation
QPSK modulation consists of Two BPSK modulation on in phase and quadrature components of the signal. The corresponding constellation is presented on figure.
1. The BER of each branch is the same as BPSK:
The symbol probability of error (SER) is the probability of either branch has a bit error:
(7)
Since the symbol energy is split between the two in-phase and quadrature components, 
and we have:
(8)
We can use the union bound to give an upper bound for SER of QPSK. Regarding fig-1, condition that the symbol zero is sent, the probability of error is bounded byy the sum of probabilities of 0 
1, 0 
2 and 0 
3. We can write
Since 
we can write:
Using the tight approximation of Q function for z>>0:
We obtain
Using Gray coding and assuming that for high signal to noise ratio the errors occur only for the nearest neighbor, 
can be approximated from 
by 
Frequency Hopping Spread Spectrum (FHSS) systems
The frequencies of the data are hopped from one to another in order to provide a secure transmission. The amount of time spent on each frequency hop is called as Dwell time.
Fig.: FHSS
Direct Sequence Spread Spectrum
DSSS is a spread spectrum modulation technique used for digital signal transmission over airwaves. It was originally developed for military use, and employed difficult-to-detect wideband signals to resist jamming attempts.
It is also being developed for commercial purposes in local and wireless networks.
The stream of information in DSSS is divided into small pieces, each associated with a frequency channel across spectrums. Data signals at transmission points are combined with a higher data rate bit sequence, which divides data based on a spreading ratio. The chipping code in a DSSS is a redundant bit pattern associated with each bit transmitted.
This helps to increase the signal's resistance to interference. If any bits are damaged during transmission, the original data can be recovered due to the redundancy of transmission.
The entire process is performed by multiplying a radio frequency carrier and a pseudo-noise (PN) digital signal. The PN code is modulated onto an information signal using several modulation techniques such as quadrature phase-shift keying (QPSK), binary phase-shift keying (BPSK), etc. A doubly-balanced mixer then multiplies the PN modulated information signal and the RF carrier. Thus, the TF signal is replaced with a bandwidth signal that has a spectral equivalent of the noise signal. The demodulation process mixes or multiplies the PN modulated carrier wave with the incoming RF signal. The result produced is a signal with a maximum value when two signals are correlated. Such a signal is then sent to a BPSK demodulator. Although these signals appear to be noisy in the frequency domain, bandwidth provided by the PN code permits the signal power to drop below the noise threshold without any loss of information.
Fig.: DSSS
Key Takeaways
1. Direct Sequence Spread Spectrum (DSSS) using BPSK modulation, so the first reversing switch introduces 180-degree phase reversals according to a pseudo-random code.
2. DSSS is a spread spectrum modulation technique used for digital signal transmission over airwaves. It was originally developed for military use, and employed difficult-to-detect wideband signals to resist jamming attempts.
Reference:
1. P Ram krishna Rao, Digital Communication, Mc Graw Hill Publication
2. Ha Nguyen, Ed Shwedyk, ―A First Course in Digital Communication‖, Cambridge
University Press.
3. B P Lathi, Zhi Ding ―Modern Analog and Digital Communication System‖, Oxford
University Press, Fourth Edition.
4. Bernard Sklar, Prabitra Kumar Ray, ―Digital Communications Fundamentals and
Applications‖ Second Edition, Pearson Education
5. Taub, Schilling, ―Principles of Communication System‖, Fourth Edition, McGraw Hill.
