In my previous tutorial, I gave a brief idea about the fundamentals of signals and their classification. Now we are going to take a step further in this direction. To do the processing part we first need to understand discrete-time signals, classification and their operations. In this tutorial major emphasis will be given on Discrete-time signals and discrete-time systems. What we are going to learn in this tutorial:-Sampling, Discrete-time Signals, Classification of Discrete-time signals, Transformation of Discrete-time signals.

First we need to understand what is a **Sampling **process? Why do we need sampling? The answer to the first question is that Sampling is a process of breakage of continuous signal to discrete signal. In a layman definition the output of system is recorded at different intervals of time, these intervals of time may not necessarily be uniform but in this series of tutorials we will limit our discussion to only **Uniform-Sampling.**

In my previous tutorial, I gave a brief idea about the fundamentals of signals and their classification. Now we are going to take a step further in this direction. To do the processing part we first need to understand discrete-time signals, classification and their operations. In this tutorial major emphasis will be given on Discrete-time signals and discrete-time systems.

What we are going to learn in this tutorial:-

{C}{C} Ø {C}{C}Sampling

{C}{C} Ø {C}{C}Discrete-time Signals

{C}{C} Ø {C}{C}Classification of Discrete-time signals

{C}{C} Ø {C}{C}Transformation of Discrete-time signals

__Sampling __

First we need to understand what is a **Sampling **process? Why do we need sampling? The answer to the first question is that Sampling is a process of breakage of continuous signal to discrete signal. In a layman definition the output of system is recorded at different intervals of time, these intervals of time may not necessarily be uniform but in this series of tutorials we will limit our discussion to only **Uniform-Sampling**.

In the above picture shown the continuous signal S(t) is being sampled at different moments of time, let the ith value be S_{i}(t) then the set of values of S_{i}(t) from i=0 to n are called the samples of S(t). Time interval between two consecutive sampling intervals is called **Sampling period **or **Sample interval**. If the time interval between two consecutive sampling intervals is uniform and equals to T,

**f _{s}= 1/T **,

**where f**

_{s }is the sampling frequency.

__Discrete-time Sinusoids__** :-**

Discrete-time sinusoids are a very important type of signal which is to be studied under Digital Signal Processing. So, since now we have a brief idea about sampling, we will be discussing about those signals and then we will get to the Sampling Theorem. A discrete-time sinusoidal signal may be expressed as,

**x(n)= Acos(w _{0}n + Ø) , **-? < n < +?

Where n is an integer, w_{0} is the angular frequency and is also equal to 2?f_{0}, where f_{0} is the frequency and Ø is the phase of the signal. Discrete-time sinusoids are periodic only if the frequency of the signal is a rational number. We need to prove this statement,

x(n+N) = x(n)

or, cos[2(pi)f_{0}(n+N) + Ø] = cos[2(pi)_{0}(n) + Ø]

or, 2(pi)f_{0}N=2k(pi)

or, f_{0} =k/N

Also, there is one more interesting properties of Discrete-time sinusoids. Discrete-time signals whose frequencies are separated by an integral multiple of 2? are identical. Let there be another signal x_{2}(t) which differs from the previous signal with a phase difference of 2?, then the x_{2}(t) can be written as,

x_{2}(t) = Acos[(w_{0 }+2(pi))n + Ø]= Acos(w_{0 }n+2(pi)n + Ø)=** **Acos(w_{0}n + Ø)

which is equal to the former signal x(t). This can be interpreted as if a signal has a frequency |f| > w. Then this signal can be said to be identical to another signal obtained from a signal with frequency |?| < ?. Thus we can say that,

|w|<=pi

Or |f| <= ½

All the frequencies in the above specified range are regarded as unique and all the other frequencies are said to be ‘aliases’.

Now it’s high time to answer the second question regarding the need of sampling, the fact that most of the signals in nature are analog caters to the need of sampling and since in my previous tutorial I have made clear benefits of Digital Signal Processing over Analog Signal Processing, to obtain Discrete-time signals we have to do sampling of the Analog signals. Now let x_{a}(t) be an analog signal and x_{a}(t) with frequency F sampled periodically to obtain x_{a}(nT) at a frequency 1/T. Then,

Ananlog signal, x_{a}(t) = A cos(2(pi)Ft + Ø)

Discrete-time signal, x_{a}(nT) = A cos(2(pi)FnT + Ø)

Or, x_{a}(nT) = A cos(2(pi)Fn/F_{s} + Ø)

As we have discussed above in the discrete-time sinusoids |f| <= ½, Thus we conclude that,

|F/F_{s}| <= ½

Or, |F| <= F_{s}/2

Thus we can clearly see that if the max. frequency of the signal is F_{max} then the sampling frequency, F_{s} must be greater than twice F_{max}. This is also known as Nyquist Rate of sampling. This is always to be remembered that F_{s} must be twice the “max.” frequency component of the signal not just any frequency of the signal otherwise it will induce attenuation and signal distortion.

**Sampling Theorem:** If the highest frequency contained in any analog signal x_{a}(t) is F_{max}=B and sampling is done at a frequency F_{s} > 2B, then x_{a}(t) can be exactly recovered from its samples using the interpolation function,

** **G(t) = sin (2wBt)/2(pi)Bt

Thus,

__Discrete-time Signals__

A function(or sequence) x(n) is said to be Discrete-time signal if the independent variable assumes integral values and carries some information .The function is not defined in the time instants between the 2 samples, it should not be takes as zero, which is a general mis-conception among the students.

Some basic or** elementary** discrete-time signals or sequences are:-

{C}{C} 1. {C}{C}**Unit sample sequence** – Represented by ?(n) and follows the following relation, given below. This signal is also referred to as Impulse.

{C}{C} 2. {C}{C}**Unit-step sequence** – Represented by u(n) and follows the following relation, given below.

{C}{C} 3. {C}{C}**Unit ramp signal** – represented by u_{r}(n) and follows the following relation, given below.

{C}{C} 4. {C}{C}**Exponential signal **– represented by x(n) and follows the following relation, given below.

x(n)= a^{n} , for all n

Where a is a parameter and the graph of x(n) could be of many types depending on the values of a. It is your task to figure out the different possibilities of x(n) depending on different values of a.

__Classification of Discrete-time signals__

**Energy and power signals: **Let there be a signal x(n). Then the power, P and energy, E of the signal is given as,

** **

Now, if the signal or sequence is having a finite energy i.e. E < ?. Then it is said to be energy signal and if the power of signal is finite i.e. P E < ?, then the signal is a power signal. There are some points to be remembered in this scenario that a signal can never be both Power and Energy signal. But a signal can be neither of both i.e. if the signal is having infinite power as well as infinite energy. If the energy of a signal is calculated over a period then it is represented as E_{N} ,

**Periodic and Non-periodic signals:** If the signal or sequence repeats itself after a fixed period of time then the signal is said to be periodic in nature with a fundamental period of N, if N is the smallest interval after which it repeats itself. Signals otherwise are non-periodic in nature, i.e. if,

For all n if, x(n+N)=x(n) , then x(n) is a periodic signal.

**Symmetric(even) and anti-symmetric(odd) signal: **A signal x(n) can be said to be symmetric and anti-symmetric if they follow the following relationship respectively,

x(-n) = x(n) {symmetric}

x(-n) = -x(n) {anti-symmetric}

Every signal can be expressed as a combination of even and odd signal, where the even component and the odd component are obtained by the following relationships,

x_{e}(n) = ½ [x(n) + x(-n)]

x_{o}(n) = ½ [x(n) – x(-n)]

The two components above follows their respective conditions of symmetricity and anti-symmetricity. And now the signal can be expressed as a sum of the even and odd components as,

x(n) = x_{e}(n) + x_{o}(n)

__Transformation of independent variable__**:-**

In further series of tutorials we will see that a discrete-time system employs some transformation of original excitation in the terms of independent variable to obtain desired response. So, here we are going to learn about some transformation that can be done on the independent variable.

{C}{C} · {C}{C}**Shifting:** A signal x(n) can be shifted in time domain by replacing the n with n-k, where k is an integer and depending on the sign of k, the signal will delay or advance in the time domain.

x(n) => x(n-k)

Then if k is positive then the signal will be delayed in time domain and if it were starting at n=0 then now it will start at n=k_{th} interval of time. Similarly if k is –ve then the signal will be advanced in time by |k| units. Although it should be taken in account that in real –time analysis of signals advancement in time has no physical significance.

{C}{C} · **Folding: **If we replace the independent variable n with –n then it is known as folding of signal or it is said to be reflection of the signal about the origin n=0.

There is a general misconception among the students about commutative properties of shifting and folding but it must be kept in mind that they are not commutative in nature, i.e. folding of delayed signal is not same as delaying of folded signals.

For example, let a signal x(n),

**first case: **x(n) is first folded and then delayed by k units, So, folding will result in x(-n) and then delaying of same folded signal will result in x{-(n-k)} = x(-n+k).

**second case:** x(n) is first delayed by k units and then folded, delaying will result in x(n-k) and then application of folding will cause the signal to be transformed into x(-n-k).

So, we can see that in both the cases the signals obtained are not same.

{C}{C} · {C}{C}**Scaling:** The third transformation revolves around the replacement of n with µn and this is known as time scaling of the signal.

In the next tutorial we are going to learn about discrete-time systems, their representations and their classification. In this series of tutorials we will be focusing on LTI(linear-time invariant) system, so there is a need to have some basic idea about discrete-time systems to further dive into the details of LTI systems.

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