Audio Signal dB levels Matlab

The simplest way would be to simply feed it through any logic IC which features Schmitt Triggering.

A Schmitt trigger has what is called input hysteresis.

This includes many of the 7400 series of logic ICs. There is a list Here, just Ctrl+F for "schmitt".

Alternatively, if you really want to do it yourself, it's pretty simple to achieve with an Op-Amp.

The way this circuit works is fairly simple.

• Take the above schematic, and imagine the "input" pin is at -V.
• Since the voltage at the "+" input is less then the voltage at the "-" input, the value of "Output" will be Low (in this case, approximately -V).
• Now, "Input" begins to rise. However, notice that there are two resistors. To cause the output to change to High, the voltage at the "+" input has to increase until i is greater then the voltage at the "-" input. This means that the value of input has to increase above the value of the negative input by an amount determined by the ratio of the resistor divider.
• Now, as soon as the "+" input is higher then the value of the "-" input, the value of output will go High (in this case, it will be approximately the value of +V).
• Since the output is also fed into the input through the resistor divider, once the output has gone positive, the threshold to cause the output to go negative changes as a function of the resistor divider.
• When the Voltage at input decreases, the same happens in reverse: Because the end of the voltage divider is now at +V, the value of input has to go below the value of the - input to cause the output to become low.

All together, this is called positive feedback (as you can see, the output is connected to the positive input. As a result, the transition threshold is affected by both the output state and the input state

The term for this, e.g. the output as both a function of the input and the previous output is Hysteresis.
When hysteresis is implemented in a circuit like this, it is called input hysteresis.

What you have appears to be a Fourier series pair, over a very small number of samples.

Probably, by looking at the definition of the Fourier series (which may be different depending on the conventions you choose), and maybe a table of results in a mathematical handbook, you can solve this by inspection.

I'll call \$x_2[n]\$ the "time domain" signal and a[k] the "frequency domain" signal.

For example, when \$N_1 \ge 2\$ then you have a pure complex exponential in the frequency domain. Therefore you know you'll have a delta-function in the time domain.

If you considered the case \$N_1 = 0\$, you'd have a delta function in the frequency domain and you'd get a pure complex exponential in the time domain.

The case \$N_1=1\$ is slightly more complicated -- that gives a complex exponential windowed by a boxcar in the frequency domain. So you know you have a sinc() (possibly offset in time) in the time domain.

But your concern about the effort involved in "writing the full expression in paper" indicates you already know how to solve the problem and you are hoping there is a shortcut or easier solution. To me, working out these Fourier relationships on paper (carefully enough to be sure of the answer) would probably be more effort than just doing the calculation in Matlab.