What are Digital Signals?

Digital Signal processing

We defined Digital signals as a sequence of values that could be complex or integers. Meanwhile, sequences are a combination of odd and even sequence.

Digital Signals: In one of our articles, we asked a question with regard to the reason why Digital signal processing is selected. Why can we not use Analog signal processing as most of the signals in practice are analog?.

Digital signal processing should be a natural choice if the signals are digital. For instance the rainfall data in your country over the last 20 years on which is based on the prediction for the next year. Predictions fail sometimes e.g data is collected for the year 1980 and then 1981 and so on which is a discrete-time signal.

If the data is to be processed digitally then the amplitude will have to be coded as well. So the natural choice would be Digital signal processing which is what is done. This data is fed with many other parameters through a supercomputer. Which basically does Digital signal processing – multiplication, addition and delay (recalling of past signals). Even if the signal is analog, Digital signals are still preferred and the reasons are many.

Digital signals have many advantages compared to Analog signals such as linear phase( which is a dominant advantage). Therefore there is no delay distortion in processing signals. We also treated the disadvantages as Digital signal processing doesn’t have only advantages. Where we discussed that the processing frequency is limited to 1.5GHz which is half of the sampling frequency of 3GHz according to the current data, as well as other disadvantages.

What are Digital Signals?

We defined Digital signals as a sequence of values that could be complex or integers. Meanwhile, sequences are a combination of odd and even sequence. However, on Digital signal processing, we talked about bounded sequences, absolutely summable sequence, square summable sequence(finite energy sequence) and average power. Then we introduced the elementary digital signals δ[n] and u[n]. We stated that u[n] can be written in terms of δ[n] as

u[n]=\sum _{ -∞ }^{ ∞ }{ δ[n-k } ]

and δ[n] in terms of u[n] as δ[n] = u[n] -u[n-1]

Also, any sequence can be written in terms of u[n] and δ[n]. Let us suppose we have a sequence;

x[n] = {0,1,2,3}

What are Digital Signals

So x[n] is expressed in terms u[n] and δ[n] as x[n] = n[ u[n]−u[n−4] ]and x[n] = δ[n−1] + 2δ[n−2] + 3δ[n−3]

Digital Signals: Exponential Sequence

One sequence that is often used is the exponential sequence of the form {Aα}^{n} . In other words, the sequence is { {Aα} , {Aα}^{2} , {Aα}^{3} …} if it starts at n = 0 → ∞. While if it start at n = −∞ → ∞ then the sequence is {…, {Aα}^{-2} ,{Aα}^{-1} , {A} , {Aα} , {Aα}^{2}, …}. Now if the sequence only extends from n = 0 → ∞ , then the way to ensure this is to multiply {Aα}^{n} by u[n] i.e,

x[n] = {Aα}^{n}_{n[n]}. Which ensures it is a right-sided signal (RSS).

Question: Is this sequence a bounded sequence?

Answer: It is a bounded sequence. If |α|< 1 because α can be complex, real, positive, or negative and if |α|> 1 then the sequence is not bounded because the magnitude can increase to infinity.

Suppose we take the general sequence {Aα}^{n} then is it bounded?. Is there some condition for it to be bounded?. No, the sequence is not bounded because if the right side of the sequence is bounded such that |α|< 1 i.e, α = 0.5, then the left side is unbounded because {0.5}^{-1} is 2 and {0.5}^{-2} is 4 and so the left side goes on increasing indefinitely. Hence the signal {Aα}^{n} in general is unbounded irrespective of the values of α.

The most important signals i.e, Information is transmitted through these signal is the sinusoidal signal e.g Acos({w}_{o}n + φ). Now is this signal a bounded signal?. Yes, it is bounded by the value A for all samples of Acos({w}_{o}n + φ). Also is this a periodic signal? Yes if Acos({w}_{o}n + φ) = Acos({w}_{o}(n + N) + φ) which satisfies the condition for a periodic signal x[n] which as we know is x[n] = x[n + N].

Exponential Sequence

Mathematically,

cos({w}_{o}[n + N] + φ) = cos({w}_{o}n + φ) where r is an integer.

Then {w}_{o}N = 2πr,

So \frac{{w}_{o}}{2π} = \frac{{r}}{N}

Therefore a digital sinusoidal signal is periodic if \frac{{w}_{o}}{2π} = \frac{{r}}{N} where \frac{{r}}{N} is a rational number since r and N are both integers.

As such \frac{{w}_{o}}{2π} is not guaranteed to be a rational number for instance take cos3n ⇒{w}_{o} = 3 and so \frac{{3}}{ 2π } is not a rational number.

Therefore the digital signals are not periodic. In other words, while all analog signals are periodic, digital sinusoidal signals may not be periodic.

Let us suppose we have 3 signals,{x}_{1}{[n]},{x}_{2}{[n]},{x}_{3}{[n]} which are periodic of period{N}_{1} ,{N}_{2},{N}_{3} respectively. Let us combine these signals in their form i.e, linearly.

{x}_{4}{[n]} = {αx}_{1}{[n]} + {βx}_{2}{[n]}+ {γx}_{3}{[n]}, is {x}_{4}{[n]} periodic?

{x}_{4}{[n]} is periodic with the period {N}_{4} = the lowest common multiple (LCM) of {N}_{1}, {N}_{2} , {N}_{3}.

For instance, let us take {N}_{1} = 3, {N}_{2} = 5, {N}_{3} = 12. Then the LCM of {N}_{1}, {N}_{2}, {N}_{3} Is 60 and after 60 cycles,{x}_{1}{[n]} completes 20 cycles, {x}_{2}{[n]} completes 12 cycles and {x}_{3}{[n]} completes 5 cycles.

Therefore at period 60, each of the signals has repeated an integer number of cycles and therefore the combination is periodic with a period of 60.

Digital Signals

Let us further explain the signal Acos({w}_{o}n + φ). How does one obtain such a signal?. Suppose we have an analog signal Acos(Ωt + φ) to obtain digital signals, we sample it at a sampling period T and thus we obtain Acos(Ωnt + φ) and after A/D conversion we get Acos({w}_{o} n + φ). So what is ω?.

ω=ΩT = \frac{Ω}{{f}_{o}} = \frac{2πf }{{f}_{o}}. Where f is the frequency of the signal in hertz. Also, ω can be written as \frac{ 2πΩ}{{Ω}_{s}}.

Notice that ω is dimensionless but ω is actually expressed in radians because it is an angle (which is the length of an arc divided by the radius). So while the analog frequency is in radian per seconds i.e, Ω(rads/s). The digital frequency, ω is in radians i.e, ω(rad) and it is called ”normalized digital frequency”.

Since the allowed frequency of digital processing cannot go past 1/2 of the sampling frequency, then {ω}_{max} = π.

Therefore the range of ω is between 0 and π{0<ω<π} and for a real signal, whatever frequency there must exist the negative frequencies also, so the range of ω is −π < ω < π. This range is called the baseband or the band of vision. In Digital signal processing it is an advantage that our range of vision is limited. Whereas in analog frequency, we go from –∞ to ∞ but this whole range is compressed to −π to +π.

Operation sequence

1.Multiplication:

If a sequence x[n] is multiplied by another sequence y[n] to give a new sequence w[n]. It means the corresponding samples are multiples i.e. sample at n = 0 of x[n] is multiplied with the sample at n=0 of y[n] to give the sample at n=0 of w[n]. This operation is symbolically representation below.

What are Digital Signals

Multiplication is a non-linear operation and sometimes we resort to such operation. Finite impulse response (FIR) filtering, multiplication of two sequences are used. We multiply the given sequences by what is called a window function.

Well, suppose the two sequences are not of the same length, then how is this operation carried out?. The missing corresponding samples are replaced with zero samples i.e. we add zero samples to the sequence with fewer samples.

2. Addition:

If we add or subtract two sequences the corresponding samples are added or subtracted. W[n] = x[n] ± y[n]. it is represented symbolically below as

What are Digital Signals

It should be noted that if the two signals are added then we do not assign a sign to the symbolic diagram, but when they are subtracted we assign a negative sign. Subtraction simply means we are changing the sign bit of signal to be subtracted.

In Digital signal processing, we cannot add two numbers at a time, for instance in an accumulator, we feed the numbers in sequence and it accumulates i.e. adds the number, replaces the storage by the result and takes another number and so on. Therefore, we must exercise this discipline. For instance, in adding 3 signals, here we will use two summers to indicate the steps that are involved.

Multiplication can also be done by a scalar for instance if we multiply a sequence x[n] by a scalar α then we represent it pictorially. This is scalar multiplication.

What are Digital Signals

3. Time reversal:

Given a sequence x[n], the time index is reversed i.e. x[n] → x[−n] so if it is a right-sided signal, it flips over to a left-sided signal and vice versa.

What is Digital Signal

Now suppose one has a sequence as shown below that starts at n = −1 then what kind of step signal would we get? It is u[−n − 1]

What is Digital Signal processing

Let us note some sequence operations e.g u[n] + u[−n] is 1 everywhere except at n=0 and it can be expressed as u[n] + u[−n] = 1 + δ[n]∀n

What is Digital Signal processing

Also u[n] + u[−n − 1] = 1∀n

Delay:

In terms of hardware, it is a one-sample delay while in terms of software, it means retrieving the immediate past sample which is stored. It is symbolized by {Z}^{−1} as shown below.

x[n] → \left[ { Z }^{ -1 } \right] → x[n-1]

This process is symbolically represented by a block whose transfer function is {Z}^{−1}. Suppose we wish to retrieve x[n − N ] sample then we feed x[n] through a chain of N delays as shown below

x[n] → \xrightarrow [ 1 ]{ \left[ { Z }^{ -1 } \right] } → \xrightarrow [ 2 ]{ \left[ { Z }^{ -1 } \right] } → ...\xrightarrow [ N ]{ \left[ { Z }^{ -1 } \right] } → x[n-1]

Conclusion

Well, we have seen that we can’t transmit signals in analog form, rather these signals have to be converter into digital form. the conversion is literally done with a converter is known as Analog to Digital(A/D) converter. Meanwhile, we have seen that Digital signals are a sequence of values that could be complex or integers. We, also saw that digital signals can undergo some operational sequences such as multiplication, addition, time reversal, and delay.

Thanks for coming around and don’t forget to check out the telecommunication archive to learn more about telecommunication.

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