Digital Signal Processing Notes

[Niklas Terki]

Anythingthat carries information can be called as signal. It can also be defined as a physical quantity that varies with time, temperature, pressure or with any independent variables such as speech signal or video signal.

Although speech and video signals have the privilege to be represented in both continuous and discrete time format; under certain circumstances, they are identical. Amplitudes also show discrete characteristics. Perfect example of this is a digital signal; whose amplitude and time both are discrete.

To test a system, generally, standard or basic signals are used. These signals are the basic building blocks for many complex signals. Hence, they play a very important role in the study of signals and systems.

If we add a step signal to another step signal that is time scaled, then the result will be unity. It is a power type signal and the value of power is 0.5. The RMS (Root mean square) value is 0.707 and its average value is also 0.5

Integration of step signal results in a Ramp signal. It is represented by r(t). Ramp signal also satisfies the condition $r(t) = \int_-\infty^t U(t)dt = tU(t)$. It is neither energy nor power (NENP) type signal.

Integration of Ramp signal leads to parabolic signal. It is represented by p(t). Parabolic signal also satisfies he condition $p(t) = \int_-\infty^t r(t)dt = (t^2/2)U(t)$ . It is neither energy nor Power (NENP) type signal.

A sinusoidal AC current signal is a perfect example of Energy type signal because it is in positive half cycle in one case and then is negative in the next half cycle. Therefore, its average power becomes zero.

A lossless capacitor is also a perfect example of Energy type signal because when it is connected to a source it charges up to its optimum level and when the source is removed, it dissipates that equal amount of energy through a load and makes its average power to zero.

Now, again compare, both the equations just as we did for conjugate signals. Here, we will find that the real part is odd and the imaginary part is even. This is the condition for a signal to become conjugate anti-symmetric type.

Shifting means movement of the signal, either in time domain (around Y-axis) or in amplitude domain (around X-axis). Accordingly, we can classify the shifting into two categories named as Time shifting and Amplitude shifting, these are subsequently discussed below.

So the y-axis being same, the x- axis magnitude decreases or increases according to the sign of the constant (whether positive or negative). Therefore, scaling can also be divided into two categories as discussed below.

Multiplication of a constant with the amplitude of the signal causes amplitude scaling. Depending upon the sign of the constant, it may be either amplitude scaling or attenuation. Let us consider a square wave signal x(t) = Π(t/4).

Whenever the amplitude of a signal is multiplied by -1, then it is known as amplitude reversal. In this case, the signal produces its mirror image about X-axis. Mathematically, this can be written as;

In the case of OPAMP differentiation, this methodology is very helpful. We can easily differentiate a signal graphically rather than using the formula. However, the condition is that the signal must be either rectangular or triangular type, which happens in most cases.

The above table illustrates the condition of the signal after being differentiated. For example, a ramp signal converts into a step signal after differentiation. Similarly, a unit step signal becomes an impulse signal.

Here also, in most of the cases we can do mathematical integration and find the resulted signal but direct integration in quick succession is possible for signals which are depicted in rectangular format graphically. Like differentiation, here also, we will refer a table to get the result quickly.

Two signals can be added first, and then their convolution can be made to the third signal. This is equivalent to convolution of two signals individually with the third signal and added finally. Mathematically, this can be written as;

Some systems have feedback and some do not. Those, which do not have feedback systems, their output depends only upon the present values of the input. Past value of the data is not present at that time. These types of systems are known as static systems. It does not depend upon future values too.

Since these systems do not have any past record, so they do not have any memory also. Therefore, we say all static systems are memory-less systems. Let us take an example to understand this concept much better.

In this expression, we are dealing with sine function. The range of sine function lies within -1 to +1. So, whatever the values we substitute for x(t), we will get in between -1 to +1. Therefore, we can say it is not dependent upon any past or future values. Hence, it is a static system.

If a system depends upon the past and future value of the signal at any instant of the time then it is known as dynamic system. Unlike static systems, these are not memory less systems. They store past and future values. Therefore, they require some memory. Let us understand this theory better through some examples.

In this case if we put t = 1 in the equation, it will be converted to x(2), which is a future dependent value. Because here we are giving input as 1 but it is showing value for x(2). As it is a future dependent signal, so clearly it is a dynamic system.

In this case, whatever the value we will put it will show that time real value signal. It has no dependency on future or past values. Therefore, it is not a dynamic system rather it is a static system.

In this case, as the system is cosine function it has a certain domain of values which lies between -1 to +1. Therefore, whatever values we will put we will get the result within specified limit. Therefore, it is a static system

Previously, we saw that the system needs to be independent from the future and past values to become static. In this case, the condition is almost same with little modification. Here, for the system to be causal, it should be independent from the future values only. That means past dependency will cause no problem for the system from becoming causal.

Here, the signal is only dependent on the present values of x. For example if we substitute t = 3, the result will show for that instant of time only. Therefore, as it has no dependence on future value, we can call it a Causal system.

Here, the system depends on past values. For instance if we substitute t = 3, the expression will reduce to x(2), which is a past value against our input. At no instance, it depends upon future values. Therefore, this system is also a causal system.

In this case, the system has two parts. The part x(t), as we have discussed earlier, depends only upon the present values. So, there is no issue with it. However, if we take the case of x(t+1), it clearly depends on the future values because if we put t = 1, the expression will reduce to x(2) which is future value. Therefore, it is not causal.

We have already discussed this system in causal system too. For any input, it will reduce the system to its future value. For instance, if we put t = 2, it will reduce to x(3), which is a future value. Therefore, the system is Non-Causal.

In this system, it depends upon the present and past values of the given input. Whatever values we substitute, it will never show any future dependency. Clearly, it is not a non-causal system; rather it is a Causal system.

An anti-causal system is just a little bit modified version of a non-causal system. The system depends upon the future values of the input only. It has no dependency either on present or on the past values.

The system has two sub-functions. One sub function x(t+1) depends on the future value of the input but another sub-function x(t) depends only on the present. As the system is dependent on the present value also in addition to future value, this system is not anti-causal.

If we analyze the above system, we can see that the system depends only on the future values of the system i.e. if we put t = 0, it will reduce to x(3), which is a future value. This system is a perfect example of anti-causal system.

In this system, if we give input as zero, the output will become zero. Hence, the first condition is clearly satisfied. Again, there is no non-linear operator that has been applied on x(t). Hence, second condition is also satisfied. Therefore, the system is a linear system.

If we want to define this system, we can say that the systems, which are not linear are non-linear systems. Clearly, all the conditions, which are being violated in the linear systems, should be satisfied in this case.

In the above system, the first condition is satisfied because if we make the input zero, the output is 1. In addition, exponential non-linear operator is applied to the input. Clearly, it is a case of Non-Linear system.

The above type of system deals with both past and future values. However, if we will make its input zero, then none of its values exists. Therefore, we can say if the input is zero, then the time scaled and time shifted version of input will also be zero, which violates our first condition. Again, there is no non-linear operator present. Therefore, second condition is also violated. Clearly, this system is not a non-linear system; rather it is a linear system.

For a time variant system, also, output and input should be delayed by some time constant but the delay at the input should not reflect at the output. All time scaling cases are examples of time variant system. Similarly, when coefficient in the system relationship is a function of time, then also, the system is time variant.

A stable system satisfies the BIBO (bounded input for bounded output) condition. Here, bounded means finite in amplitude. For a stable system, output should be bounded or finite, for finite or bounded input, at every instant of time.

In the given expression, we know that sine functions have a definite boundary of values, which lies between -1 to +1. So, whatever values we will substitute at x(t), we will get the values within our boundary. Therefore, the system is stable.

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