Digital Signal Processing

Thread Starter

Vikram50517

Joined Jan 4, 2020
27
Consider a digital system wkt y=mx(n)+c(where y is the response,x(n) is the input ,n is discrete time and c is some constant)
is a linear equation but it disobeys principle of superposition and thus the system is considered non linear.does it mean that superposition principle fails to explain its linearity in this case or what??.I can't understand the intricacies of the difference these 2 (linearity as defined by superposition principle and linearity as defined by characteristics)
 

MrAl

Joined Jun 17, 2014
7,672
Consider a digital system wkt y=mx(n)+c(where y is the response,x(n) is the input ,n is discrete time and c is some constant)
is a linear equation but it disobeys principle of superposition and thus the system is considered non linear.does it mean that superposition principle fails to explain its linearity in this case or what??.I can't understand the intricacies of the difference these 2 (linearity as defined by superposition principle and linearity as defined by characteristics)
Hi,

Just a quick guess here but there is only one definition of linearity but we sometimes accept another too.

The definition of linear is really a straight line though the origin. This would be something like y=m*x.
A straight line NOT through the origin is sometimes accepted as linear, but it may not work with superposition. This would be something like y=m*x+b where b is a constant. The constant makes it strictly speaking not linear anymore.

Try some experiments see what you can find out.
 

bogosort

Joined Sep 24, 2011
465
Consider a digital system wkt y=mx(n)+c(where y is the response,x(n) is the input ,n is discrete time and c is some constant)
is a linear equation but it disobeys principle of superposition and thus the system is considered non linear.does it mean that superposition principle fails to explain its linearity in this case or what??.I can't understand the intricacies of the difference these 2 (linearity as defined by superposition principle and linearity as defined by characteristics)
The word linear is sometimes used in two incompatible ways. In the context of signals and systems (and, more generally, of mathematical spaces), linearity is a property of maps between sets. If the mapping respects addition (superposition) and scaling (homogeneity), we say that the map is linear. So, for example, some function \[ f: \mathbb{R} \to \mathbb{R} \] is linear if and only if the following conditions hold:
\[ f(a + b) = f(a) + f(b) \qquad \text{ and } \qquad f(ca) = c f(a) \qquad \forall a, b, c \in \mathbb{R} \] However, in elementary algebra we often hear the general equation of a line, \( y = mx + b \), called a "linear equation", even though y(x) is not a linear function in the sense described above. This usage of "linear" is informal and meant to be descriptive of the geometric shape it represents rather than its mathematical properties, and should not be confused with the technical definition of linearity.

Another common source of confusion is the notion of a linear system, which may be composed of nonlinear functions. So, for example, the system \( y(t) = t^2 x(t) \) has a nonlinear (square-law) component, yet the system itself is linear. The key thing to remember is that linearity (in the formal sense) is a property of maps between sets, applying to the map itself and not the elements being mapped. Functions are maps between sets of numbers, whereas systems are maps between sets of functions. In other words, a function takes a number for an input and produces a number at its output, whereas a system takes a function for an input and produces a function at its output. In the first case, the function itself must be linear because the function is the map; in the second case, the functions are the elements, not the map, and so they can be nonlinear. As long as the map itself is linear, then the system is linear, regardless of the linearity of its components.
 

Thread Starter

Vikram50517

Joined Jan 4, 2020
27
Hi,

Just a quick guess here but there is only one definition of linearity but we sometimes accept another too.

The definition of linear is really a straight line though the origin. This would be something like y=m*x.
A straight line NOT through the origin is sometimes accepted as linear, but it may not work with superposition. This would be something like y=m*x+b where b is a constant. The constant makes it strictly speaking not linear anymore.

Try some experiments see what you can find out.
Yes,Thank you very much.
 

Thread Starter

Vikram50517

Joined Jan 4, 2020
27
The word linear is sometimes used in two incompatible ways. In the context of signals and systems (and, more generally, of mathematical spaces), linearity is a property of maps between sets. If the mapping respects addition (superposition) and scaling (homogeneity), we say that the map is linear. So, for example, some function \[ f: \mathbb{R} \to \mathbb{R} \] is linear if and only if the following conditions hold:
\[ f(a + b) = f(a) + f(b) \qquad \text{ and } \qquad f(ca) = c f(a) \qquad \forall a, b, c \in \mathbb{R} \] However, in elementary algebra we often hear the general equation of a line, \( y = mx + b \), called a "linear equation", even though y(x) is not a linear function in the sense described above. This usage of "linear" is informal and meant to be descriptive of the geometric shape it represents rather than its mathematical properties, and should not be confused with the technical definition of linearity.

Another common source of confusion is the notion of a linear system, which may be composed of nonlinear functions. So, for example, the system \( y(t) = t^2 x(t) \) has a nonlinear (square-law) component, yet the system itself is linear. The key thing to remember is that linearity (in the formal sense) is a property of maps between sets, applying to the map itself and not the elements being mapped. Functions are maps between sets of numbers, whereas systems are maps between sets of functions. In other words, a function takes a number for an input and produces a number at its output, whereas a system takes a function for an input and produces a function at its output. In the first case, the function itself must be linear because the function is the map; in the second case, the functions are the elements, not the map, and so they can be nonlinear. As long as the map itself is linear, then the system is linear, regardless of the linearity of its components.
Cleared my doubts!
Thank you very much
 
Top