Here’s a system-level look at the signals and systems model of a karaoke machine — an audio playback system with a powerful speaker that allows a person to sing over recorded music. A multimedia interface includes a TV to display and update lyrics as the music progresses.

From a high-level signals and systems viewpoint, a particular design attribute of this system is that it contains a sensor, a microphone, and two audio transducers (the left and right channel speakers).

See a block diagram for this karaoke machine in the following figure.

Credit: Illustration by Mark Wickert, PhD

The signal flow through this system consists of two paths: one for the recorded music and the other for the singer’s voice that enters the microphone. The subsystems of the karaoke machine act upon the two input signal types — in this case, both random signals — to finally end up at the speakers, which convert the electrical signals to sound pressure waves that your ears can interpret.

The karaoke machine as a system has three input signals — xl(t), xr(t), and xm(t) — and two output signals — y1(t) and y2(t). The input xm(t) represents the voltage signal produced by the microphone (sensor). The outputs represent the voltage signal that drives the speakers (transducers). The system input-output equations are

The constants Gm, Gl, and Gr represent scaling factors that provide the following system needs:

• Extra gain for the microphone (a preamp)

• High power output stage so the speakers can produce the beautiful music that karaoke is known to produce

Knobs or slider controls on the user interface also alter the gain constants to help strike a balance between the recorded music and the singer levels. The system input-ouput equations tell you that the system is time-invariant for fixed gain constants, memoryless, and thus causal.

Assuming that the gain constants are finite (which in a practical system is the case), the system is also stable. Note that if you raise and lower the gain parameters (think volume control), the system becomes time-varying because the property of the system is now a function of time.

Other enhancements may be applied. For example, the subsystem formed by the microphone pathway, Gmxm(t), can be upgraded to include filtering for bass and treble tone controls. This filter is equivalent to a linear time-invariant system (LTI).

Filtering introduces memory, so the system is no longer memoryless with this enhancement; but to remain practical, the system still must remain causal. Special effects, such as reverb or echo, can be added to the microphone channel. A system model for the reverb would include a system property for the intensity of the delay factor in the reverb. An LTI system can implement the reverb.

This is just one of countless examples of how signals and systems modeling allows you to be creative and functional at the same time!