Relations between LTI system properties and the impulse response. 

The impulse response completely characterizes the input-output behavior of an LTI system. Therefore the properties of the system, such as the memory, the causality, and the stability, are related to the system’s impulse response. 

Further on, we will be exploring the relationships involved. 

Memoryless LTI system

As we know, the output of a memoryless LTI system depends only on the current input. Exploiting the commutative property of convulation, the output of a discrete-time LTI system can be expressed as follows:

Y[n] = h[n]*x[n] 
        = Σ h[k]x[n-k]

Expand the sum term by term:

y[n] = …+ h[-2] x[n+2] + h[-1] x[n+1] + h[0]x[n]+.....

For a system to be memoryless, y[n] must depend only on x[n] and therefore cannot depend on x[n-k] for k=! 0. 

Therefore, every term in the given equation must be zero, except h[0]x[n]. This condition implies that h[k]=0, for all k=! 0. 

Thus, a discrete-time LTI system is memoryless if and only if: 

H[k] = cδk. 

Where c is an arbitrary constant.

Writing the output of a continuous-time system is:

y(t) = ∫ h(T)x(t-T) dT.  

We see, analogously to discrete-time case, a continuous-time LTI system is memoryless if and only if. 

h(T) = cδk. 

For an arbitrary constant. 

The memoryless condition places severe restrictions on the form of the impulse response: All the memoryless LTI systems simply perform scalar multiplication on the input. 

Image taken from wikidot

Causal LTI systems

The output of a causal LTI system depends only on past or present values of the input. 

We write the convolution sum as:

Y[n] = …+h[-2]x[n+2] + h[-1]x[n+1] +h[0]x[n]+ ….. 

We see that past and present values of the input x[n], x[n-1], x[n-2],….. Are associated with indices k>=0 in the impulse response h[k], while future values of the input, x[n+1], x[n+2],…., are associated with indices k<0. In order then, for y[n] to depend only on past or present values of the input, we require that h[k] = 0 for k < 0, therefore, for a discrete-time causal LTI system, 

H[k] = 0 for k<0. 

And the convolution sum takes the new form:

y[n] = Σ h[k]x[n-k]. 

The causality condition for a continuous-time system follows in an analogous manner from the convolution integral:

y(t) = ∫ h(T)x(t-T) dT.  

The causality condition for a continuous-time LTI system has an impulse response that satisfies the condition. 

h(T) = 0 for T <0. 

The output of a continuous-time casual LTI system is thus expressed as the convulation integral. 

 y(t) =  ∫ h(T)x(t-T) dT. 

The causality condition is intuitively satisfying. Recall the impulse response is the output of a system in response to a unit-strength impulse input applied at time t = 0. Note that causal systems are nonanticipatory; that is, they cannot generate an output before the input is applied. Requiring the impulse response to be zero for negative time is equivalent to saying that the system cannot respond to be zero for negative time is equivalent to saying that the system cannot respond with an output prior to application of the impulse. 

Image taken from researchgate. 

Stable LTI systems

The system is bounded input-bounded output or BIBO stable if the output is guaranteed to be bounded for every bounded input. Formally, if the input to a stable discrete-time system satisfies |x[n] <= Mx <= infinity, then the output must satisfy |y[n]| <= My <= infinity. 

Invertible systems and deconvulation

A system is invertible if the input to the system can be recovered from the output except for a constant scale factor. This requirement implies the existence of an inverse system that takes the output of the original system as its input and produces the input of the original system. 

We should limit ourselves here to a consideration of the inverse systems that are LTI. 

The process of recovering x(t) from h(t) * x(t) is termed as deconvulation, as it corresponds to reversing or undoing the convolution operation. An inverse system performs deconvulation. 

Deconvulation problems and inverse systems play an important role in many signal processing and system applications. A common problem is that of reversing or equalizing the distortion introduced by a nonideal system. For example, consider the use of a high-speed modem to communicate over telephone lines. Distortion introduced by the telephone channel places severe restrictions on the rate at which information can be transmitted, therefore an equalizer is incorporated into the modem. 

It reverses the distortion and permits much higher data rates to be achieved. In this case, the equalizer represents an inverse system for the telephone channel. 

Reference

Relations between LTI system properties and the impulse response