Properties of Discrete-Time System

1- System with Memory

A system that has memory if its output at time 𝑌0 𝑛 depends on input values other than 𝑋0 𝑛 . Otherwise the system is memoryless. For a discrete signal x[n], time is represented by the discrete increment variable n. An example of a simple memoryless discrete-time system is the equation

𝑦[𝑛] = 5𝑥[𝑛] A memoryless system is also called a static system. A system with memory is also called a dynamic system. An example of a system with memory is the Euler integrator of

𝑦[𝑛] = 𝑦[𝑛 − 1] + 𝐻𝑥[𝑛 − 1]

This equation can be represented as: 

And we see that the output depends on all past values of the input. A second example of a discrete system with memory is one whose output is the average of the last two values of the input. The difference equation describing this system is:

System with Inevitability

A system is said to be invertible if distinct inputs result in distinct outputs. A second definition of inevitability is that the input of an invertible system can be determined from its output. For example, the memoryless system described by

 not invertible The inputs of +2 and -2 produce the same output of +2

Inverse System

Inverse of a System The inverse of a system T is a second system 𝑇𝑖 that, when cascaded with T, yields the identity system.

The identity system is defined by the equation 𝑌[𝑛]=𝑋[𝑛] Consider the two systems of the figure below. System 𝑇𝑖 is the inverse of system T if

Casual System

A system is causal if the output at any time is dependent on the input only at the present time and in the past.

We have defined the unit delay as a system with an input of x[n] and an output of

𝑋[𝑛−1].

3

An example of a non-causal system is the unit advance, which has an input of x[n] and an output of 𝑋[𝑛+1]

Another example of a non-causal system is an averaging system, given by

Which requires us to know a future value, 𝑋[𝑛+1] of the input signal in order to calculate the current value, y[n], of the output signal.

We denote the unit advance with the symbol 𝐷−1

We realize it by first delaying a signal and then advancing it. However, we cannot advance a signal more than it has been delayed. Although this system may appear to have no application, the procedure is used in filtering signals “off line,” or in non-real time. If we store a signal in computer memory, we know “future” values of the signal relative to the value that

Stability

BIBO Stability: A system is stable if the output remains bounded for any bounded input. This is the bounded-input bounded-output (BIBO) definition of stability. By definition, a signal x[n] is bounded if there exists a number M such that

Hence, a system is bounded-input bounded-output stable if, for a number R,

For all x[n] such that (1) is satisfied. To determine BIBO stability, R [in general, a function of M in (1)] must be found such that (2) is satisfied. Note that the Euler integrator of

The BIBO is stable; if the signal to be integrated has a constant value of unity, the output increases without limit as n increases.

Time Invariance

Time-invariant System: A system is said to be time invariant if a time shift in the input results only in the same time shift in the output. In this definition, the discrete increment n represents time. For a time-invariant system for which the input x[n] produces the output y[n], the input produces

A test for time invariance is given by:

Linearity

The property of linearity is one of the most important properties that we consider. Once again, we define the system input signal to be x[n] and the output signal to be y[n].

These two criteria can be combined to yield the principle of superposition. A system satisfies the principle of superposition if

Where and are arbitrary constants. A system is linear if it satisfies the principle of superposition. No physical system is linear under all operating conditions. However, a physical system can be tested with the use of (1) to determine ranges of operation for which the system is approximately linear. An example of a linear operation (system) is that of multiplication by a constant K, described by an example of a nonlinear system is the operation of squaring a signal,

A linear time-invariant (LTI) system is a linear system that is also time invariant. LTI systems, for both continuous-time and discrete-time systems, are emphasized in this book. An important class of LTI discrete-time systems are those that are modeled by linear difference equations with constant coefficients.

In this equation, x[n] is the input, y[n] is the output, and the numerical-integration increment H is constant. The general forms of an nth-order linear difference equation with constant coefficients are given by:

This difference equation is said to be of order N. The second version of this general equation is obtained by replacing n with (𝑛+𝑁).

Digital Audio Signals in matlab

MATLAB Audio Functions

There are several functions in MATLAB for working with audio. These functions include audioread, audiowrite, audioinfo, and sound

audioread

The audioread function can be used to import an audio le into MATLAB. The samples in the audio le are stored in a column vector. Additionally, the sample rate can be read from the audio le and assigned to a scalar variable. The syntax for using the function is the following: [y,Fs] = audioread(filename). 

It is necessary for the variable, filename, to be a string with the name of an audio le stored on the computer. Types of supported les for the audioread function are .wav, . ac, .mp3, .ogg, .aiff, and several others. The le extension should be included in the filename. 

In Example 3.1, the syntax for the audioread function is demonstrated. The result of the example is an array, sw, containing the samples of the signal and a variable, Fs, with the sampling rate of the le stored on the computer. The variables are printed to the Command Window to view. Throughout the book, sound les referenced in examples are included in the book’s Additional Content (www.hackaudio.com). 

Examples: Execute the following commands in MATLAB. 

>> fn = 'sw440Hz.wav'; 

>> [sw,Fs] = audioread(fn)

sound

The sound function can be used to play a MATLAB array of audio samples through the computer’s output device. It is necessary to specify the appropriate sampling rate, Fs, for playback. The syntax for using the function is the following: sound(y,Fs). The commands to import a sound le and play it are demonstrated in Example 3.2. 

Examples: Execute the following commands in MATLAB. 

>> fn = 'sw440Hz.wav'; 

>> [sw,Fs] = audioread(fn); 

>> sound(sw,Fs); 

audiowrite

The audiowrite function can be used to create an audio le from a MATLAB array, which is then stored on the computer. The name of the le is specied by a string, lename. Folder location can also be referenced in the name of the le. The name of the MATLAB array and 

the appropriate sampling rate are specied by the next two variables, respectively. The syntax for using the function is the following: audiowrite(filename,y,Fs). Executing the commands in Example 3.3 creates a new sound le saved to the hard drive of the computer. 

Examples: Execute the following commands in MATLAB. 

>> fnIn = 'sw440Hz.wav'; 

>> [sw,Fs] = audioread(fnIn); 

>> fnOut = 'testSignal.wav'; 

>> audiowrite(fnOut,sw,Fs);

audioinfo

The audioinfo function can be used to nd out information about the contents of an audio le. This information includes the name of the le, whether the le is mono or stereo, sampling rate, number of bits per sample, as well as other information. The syntax for using the function is the following: info = audioinfo(filename). 

Examples: Execute the following commands in MATLAB. 

>> fn = 'sw440Hz.wav'; 

>> info = audioinfo(fn)