## Discrete time signals in Discrete MathematicsThe signals will be known as discrete-time signals if they are defined only at discrete intervals of time. With the help of symbol x[n], we can indicate the discrete-time signals, where n is used to indicate the independent variable in the time domain. In other words, we can use the signals of numbers to indicate the discrete-time signals, which are described as follows: Here n is used to indicate the integer, and x[n] is used to indicate the nth sample in the sequence. With the help of sampling continuous-time signals, we can often get discrete-time signals. In this type of case, the nth sample of the sequence and the analog signal's value x x[n] = x After that, the Because of the above reason, although x[n] is strictly known as an nth number in the sequence, but we can often refer to it as an nth In the graphical form, we can often depict the discrete-time signals in the following way: With the help of MATLAB function stem, we can plot the discrete-time signals. Here the value of x[n] is ## Representation of Discrete-time signals- Graphical representation
- Functional representation
- Tabular representation
- Sequence representation
Here we will assume a discrete-time signal x(n) with some values, which are described as follows: x(0) = -1, x(-1) = 0, x(-2) = 3, x(-3) = -2, x(3) = 1, x(2) = 3, x(1) = 2 We can represent the discrete-time signal graphically in the following way:
The magnitude of the signal will be written against the values of n in case of the functional representation of a discrete-time signal. Therefore, in the following way, we can represent the above discrete time signal x(n) with the help of functional representation like this:
In case of a tabular representation of discrete-time signal, we use the table to represent the sampling instant n and the magnitude of discrete-time signal at the corresponding sampling instant. In the following way, we can represent the above discrete time signal x(n) with the help of a tabular form like this:
In the form of sequence representation, we can represent the discrete-time signal x[n] in the following way: Here The arrow mask (↑) is used to indicate the term corresponding to n = 0. If there is a case in which sequence representation of a discrete-time signal does not contain any arrow, then the first term of this sequence will correspond to n = 0.
With the help of adding the corresponding element of the sequence, we can get the sum of two discrete-time sequences, which is described as follows: With the help of multiplying the corresponding element of the sequence, we can get the product of two discrete-time sequences, which is described as follows: With the help of multiplying each element of the sequence and a constant, we can get the product of a sequence and a constant k, which is described as follows: {C There are several ways in which we can manipulate the sequences. We can define the sum and product of two sequences, x[n] and y[n] with the help of respectively doing sample by sample sum and product. If we are multiplying x[n] by a, then it is similar to multiplying each sample value by a. Here y[n] will be known as the Y[n] = x[n-n Here n ## Unit sample SequenceThe unit sample sequence can be defined in the following way: The graphical representation of unit sample sequence is described as follows: We can also call this type of sequence as an The impulse sequence is used to have an important aspect, i.e., we can represent the arbitrary sequence in the form of a sum of scale and delayed impulse.
x[n] = a In general, we can use the following way to express any sequence: The unit sequence can be defined in the following way: The graphical representation to show unit sequence is described as follows: In the following way, the unit step can be related to the impulse: There is another way in which we can define it, which is described as follows: Conversely, we can express the unit sample sequence in the form of first backward difference of the unit step sequence like this: ## Exponential SequenceWe can analyze and represent the discrete-time systems with the help of x[n] = Aα If there are two real numbers, A and If there is a case where A x[n] = A cos(w0n + φ) for all n, Here A and φ are used to indicate the real constants. The graphical representation of the sinusoidal sequence is described as follows: In the following way, we can express the exponential sequence A ^{φ} like this:So the real and imaginary parts are used to indicate the exponentially weighted sinusoids. The sequence will be known as the complex exponential sequence if | x[n] = |A| e So this complex sinusoid is used to have the w The index n is always used to indicate an integer. Because of this, we can get some important differences between the properties of continuous-time complex exponentials and discrete-time complex exponential, which is described as follows: For this, we will assume a complex exponential with frequency (w0 + 2π) like this: x[n] = Ae So in conclusion, we can say that the sequence for complex exponential, which has frequency w x[n] = A cos[(w The complex exponential sequence and sinusoidal sequence are both always periodic in the case of continuous time. If the discrete-time sequences will be periodic if it contains the following: Thus, the discrete-time sinusoid will only have period if it contains the following: A cos[(w This sequence needs the following: We need the same conditions if we need to show that the complex exponential sequence C e When we combine the above described two factors, then we will reach a conclusion that says that there will be N distinguishable frequencies only for which the corresponding sequences are periodic with period N. This type of set is described as follows: w Additionally, in the process of discrete-time sequences, we should modify the interpretation of high and low frequencies. If there is a discrete-time sinusoidal sequence x[n] = A cos[(w If there is a sequence that corresponds to w Next TopicM-array Tree in Discrete Mathematics |