Before we take a lot at the various classification of the digital system, its best we know what digital systems are. This will build our understanding when we start talking about the classification.

Before we take a lot at the various classification of the digital system, its best we know what digital systems are. This will build our understanding when we start talking about the classification.

A digital system handles *digital signals* and operates on digital signals to produce another digital signal which in some ways is better than the previous digital signals. Let us take an example, consider the digital system shown below.

Which results in a difference equation given as;

y[n] ={b}_{o}x[n] + {b}_{1} x[n − 1] + {d}_{1} y[n − 1].

It is a differential equation because we have x[n], x[n − 1] also y[n] and y[n − 1] which is an exact counterpart of differential equation in analog signal processing. However, any dynamic system will be described by a difference equation and any difference equation can always be represented by a schematic diagram shown above.

The diagram above represents hardware as well as software. In terms of hardware, we need delay elements, multipliers (accumulators), summers (also accumulators). While in terms of the software it shows what is multiplied by what and which signal is to be retrieved. → *Classification of Digital Systems*

Table of Contents

## Sampling Process

If we have an analog signal cos({Ω}_{o}t + φ). We get a digital signal by sampling at T and A/D conversion (though we do not show A/D because of the mathematical techniques for the discrete-time signal {x}_{a} [nT] and the digital signal x[n] are the same). So:

Where ω = \frac{{2π}{f}_{o}}{{f}_{s}}. We stated earlier that {f}_{o} ≤ \frac{{f}_{s}}{2}. So, what happens if this is not the case? Let us say we have three signals cos6πt,cos14πt, and cos26πt. Each of frequencies 3Hz, 7Hz, and 13Hz respectively. Let us sample these signals at 0.1s (i.e.{f}_{s} = 10Hz).

Note 10Hz is not the right sampling frequency for cos14πt and cos26πt signals because {f}_{o} ≤ \frac{{f}_{s}}{2}. But let us see what happens. So, sampling at 0.1s gives cos(0.6πn), cos(1.4πn) and cos(2.6πn) since {ω}_{o} =\frac{{2π}{f}_{o}}{{f}_{s}} .

Now let us examine the digital signals cos(1.4πn) and cos(2.6πn) closely, we observe that;

cos(1.4πn) = cos(2πn − 0.6πn) = cos(2πn)cos(0.6πn) + sin(2πn)sin(0.6πn) = cos(0.6πn)

Also

cos(2.6πn) = cos(2πn + 0.6πn) = cos(2πn)cos(0.6πn) − sin(2πn)sin(0.6πn) = cos(0.6πn)

In other words, the three signals become indistinguishable. This is what inadequate sampling does. Firstly, the 7Hz signal poses as a 3Hz signal, the same applies to the 13Hz signal. So higher frequencies pose as low frequencies and this process is known as ”aliasing”. → *Classification of Digital Systems*

## Aliasing

This term is usually used in criminal language due to a terrorist called an alias. So, it is the same idea that is taken in this analogy because the high frequencies pose as low frequencies. Also, the distortion due to this factor is known as ”aliasing distortion”. The solution to aliasing distortion is to make sure {w}_{o} ≤ π or {f}_{o} ≤ \frac{{f}_{s}}{2}

For example; suppose T = 0.02sec ⇒ fs = 50Hz then the signals will be cos(0.12πn),cos(0.28πn) and cos(0.52πn) respectively. Meanwhile, each ωo is less than π. So each signal is distinct. Now, what is the lowest sampling frequency one can take for these three signals?. It is 26Hz (since the highest frequency is 13Hz then twice of 13Hz gives the sampling frequency). This essentially is the sampling theorem.

*Classification of Digital Systems *

## Examples of Digital Systems

Well, before we take a look at the classification of the digital system, lets us take a few examples of a digital system. Moreover, there are various examples of digital systems as listed below. Meanwhile, we also have in-depth knowledge of what they are.

- Accumulator
- Up sampler
- Down sampler
- M-point moving average system

**Accumulator:**

Well, an accumulator simply adds up digital signals. So, it is given as

**y[n]=\sum _{ l=-∞ }^{ n }{ x[l] } **

This shows that the system adds the previous number of signals x[n] up to the present instance is to give y[n]. The expression can be written in many different ways. Meanwhile, one way to rewrite it is; y[n] = y[n−1]+x[n]; which is a difference equation.

This equation is a recursive equation because it requires feedback y[n − 1] and whenever there is recursion there is a possibility of oscillation or instability. On the other hand, the first equation does not require a past output (non-recursive). One other way to express the first equation is

**y[n]=\sum _{ l=-∞ }^{ -1 }{ x[l] }+\sum _{ l=0 }^{ n }{ x[l]}**

**⇒y[n]=y[−1] +\sum _{ l=0 }^{ n }{ x[l] }**

Hence y[−1] is in the initial condition and we start computing at l = 0. We will see later that depending on how we express the equation for the accumulator system the characteristics are affected i.e one might be linear and the other is not.

**Up sampler:**

It is a digital system that is completely used to change the rate of sampling. For instance, for samples of 10Hz, we could change it to 20Hz using the upsampler or similarly a 40Hz signal can be downsampled to a 3Hz sampling rate. It is represented by the equation.

y[l]=\begin{cases} x[\frac { n }{ l } ]\quad n=0,\pm l,\pm 2l \quad 0\quad \quad otherwise \end{cases}Let us examine a typical signal shown below;

Suppose we have a sampling rate L=3, then y[n] is shown above.Because y[n] = x[n] if n = 0. For n = 1,y[1] = x[1/3] which does not exist. So at n=1, y[n] is zero.

Similarly at n = 2, y[2] = x[2/3] which does not exist and so it is zero.

At n = 3, y[3] = x[3/3] = x[1] which exists and the value is 2.

At n = −1, y[n] does not exist.

At n = −2, y[n] does not exist.

But at n = −3, y[n] = −1 and the rest samples of x[n] are zeros.

So what has the upsampler done? The picture of the signal has not changed but it has expanded because it has added 2 zeros in between each sample of x[n]. Therefore the upsampling factor of 3 means feeding 2 = [3 – 1] zeros i.e [L − 1] zeros between consecutive samples. We shall see later the nature of the system such that it is a time-varying system.

Therefore the nature of the signal is not altered but it is stretched.

**Down sampler**:

Similarly, we can describe the downsampler in the same sense as we described the up sample.

and y[l]=\begin{cases} x[Mn]\quad n=0,\pm M,\pm 2M \quad 0\quad \quad otherwise \end{cases}

It simply means that a downsampler simply ignores M-1 samples between 0 and M or M and 2M because y[n] exist for samples 0, ±M, ±2M and so on.

## M-point moving average system:

Hence the system sums up the present input x[n] up to the past M-1 input i.e, x[n − m + 1] and divides by the number of M inputs. This is why it is called the average but it is a moving average because as n changes, we take the previous M-1 samples. So, as we move with n the average also changes.

This is a very useful device for data smoothing, let us suppose we have a signal x[n] which comprises the true signal, s[n] and the noise, d[n] i.e x[n] = s[n] + d[n] then when we take the moving average of x[n], then the noise component d[n] diminishes. In a usual sequence, N is large and so the average of d[n] is zero. Therefore moving average system is also called the **data smoothing system**.

## Classification of Digital Systems

**1.**A digital system can be either linear or non-linear. A digital system is linear if and only if

X1,2[n] → Y1,2[n] i.e X1[n] leads to Y1[n] and X2[n] leads to Y2[n]. This implies that αx1[n] + βx2[n] should lead to αy1[n] + βy2[n].

**2.****Time invariance:**Since time has lost its significance in the digital system. It is also sometimes called**shift-invariance**. The definition is , if x[n] → y[n] then it implies x[n–{n}_{o}] should lead to y[n − {n}_{o} ] i.e x[n − {n}_{o}] → y[n–{n}_{o}].

The interpretation is that whether the input waveform is shifted or not the output is also shifted .i.e. the shift is preserved.

Therefore the shape of the input waveform is not preserved and the system is a time-varying system similarly we can show that the downsampler is also time-variant.

Let us take an example of a signal x[n] = { ↑ 1 2 3} that is upsampled by L = 2 and test for time invariance.

Now let us delay x[n] by 1 sample so that the waveform becomes x[n − 1] and then let us up sample x[n − 1] by a factor of 2 so that we get;

This shows that y[n-1] is not the same as y[n] if it is delayed by 1 sample.

**3.Causality: **

A Causal system is one that can not predict the future. Another name of a causal system is a realizable system. The definition is y[n] depends on x[n], x[n − i] and y[n − j] where i and j are positive integers, that is y[n] the present output depends on the present input x[n], past input x[n − i] or past output y[n − j]. → *Classification of Digital Systems*

Suppose we have a signal x[n + 1] which means I allow both positive and negative integers, then it will mean our system at the present moment predicts what will come one sample later and it is not permitted. We do not have a device that can advance time, we can only delay time. Let us look at the formal definition (mathematically);

*Theorem:*

*Theorem:*

A digital system is causal if and only if x1[n] ≡ x2[n] for n ≤ N, where N is an integer which implies y1[n] ≡ y2[n] for n ≤ N.

It means if n exceeds N x1[n] and x2[n] may be different then y1[n] and y2[n] may also be different. But so long as the two inputs x1[n] and x2[n] are identical what comes in future for x1 and x2 are not anticipated by the system. Let us take a simple example:

Supposed we have a system y[n] = x[n2]. Is this system causal?

No, it is not because n = 2 then y[2] = x[4] and x[4] is yet to come so the system is not causal.

Also this system is not causal because y[n] depends on x[n + 1]. Now if we cannot realize a non-causal system then **why are we considering it?** **Why are we making a distinction?** The distinction comes because non-causal systems in Digital signal processing is possible from recorded data. The geophysicist, for example, goes to the field and takes records by vibrations and he gets back to the laboratory where he analyzes it.

Now when he is analyzing for a particular output y[n] he knows what is going to come because it is a recorded data. So on recorded data non-causal signal processing is possible. Also in weather prediction non-causal digital signal processing is possible but we cannot realize a non-causal system in hardware. → *Classification of Digital Systems*

A signal x[n] is causal if and only if x[n] = 0 for n < 0. We are to start the signal somewhere and call that sample n = 0 so that the signal is causal. If we have a value at x[−1], x[0] then the signal is not causal.

**4. Stability: **

Except we need a digital oscillator, all digital systems are stable in order to be useful and the stability we bother about in Digital signal processing is that of **Bounded Input Bounded Output (BIBO)**.

** Theorem :** A digital system is stable if and only if

|x[n]| ≤ {B}_{x} < ∞ then |y[n]| ≤ {B}_{y} < ∞ where{B}_{x} and {B}_{y} are integers. For example y[n] =\frac{1}{{x}^{2}} is an unstable system.

**5. Passivity: **

It means the energy of the output cannot exceed the energy of the input. **Theorem:** A system is said to be passive if and only if

**\sum _{ n=∞ }^{ ∞ }{ { |y[n]| }^{ 2 } }≤ \sum _{ n=∞ }^{ ∞ }{ { |y[n]| }^{ 2 } } **

In order words the system does not generate energy, for instance in *RC networks* we have a passive network but if combined with op-amps or transistors the network becomes active. If the above theorem is satisfied at the equality sign then the system is lossless.

Well, that's all we have for you as regards the classification of digital systems. I hope it helped out your search.

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

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