For analog signals, most of the processing operations are performed in the time domain, on the other side, for discrete-time signals the time and frequency domain operations are employed.

# Simple Time-Domain Operations

The signal processing operations are building blocks that will allow to design a more complex set of processing tasks. These operations are detailed as follows:

#### Scaling

The multiplication of the signal by a positive/negative constant. For analog signals, this operation is called amplification if the multiplier is greater than one, otherwise is called attenuation. More formally, if $$x(t)$$ is the analog signal, the scaling operation generates a $$y(t) = \alpha x(t)$$.

#### Delay

Generates a replica signal delayed in respect to the original. For an analog signal $$x(t)$$, a delayed signal will be $$y(t) = x(t - t_0)$$ where $$t_0$$ is the amount of time units in which the signal is delayed. If $$t_0$$ is positive, then this is a delay operation, otherwise, is an advance operation.

Two or more signals can be added to generate a new one. Lets say $$y(t) = x_1(t) + x_2(t) - x_3(t)$$ is a signal formed by the addition (and substraction) of three signals.

#### Product

Another elementary operation is the product of two signals, for example $$y(t) = x_1(t) x_2(t)$$

#### Integration and differentiation.

The integration of a signal $$x(t)$$ will generate a signal $$y(t) = \int_{-\infty}^{t} x(\tau) \mathrm{d}\tau$$, while the differentiation is defined as $$w(t) = \frac{\mathrm{d}x(t)}{\mathrm{d}t}$$.

It should be noticed that the first three elementary operations, scaling, delay and addition, can be also applied to discrete-time domain signals. However, integration and differentiation are used mostly for discrete-time domain signals.

# Filtering

The main objective of filtering is to alter the spectrum according to some provided specifications. For example, a filter can be designed to allow some specific frequency components to stay in the signal, while others can be blocked. A passband is the range of frequencies allowed to pass through the filter, whereas a stopband is the range of frequencies blocked by the filter.

More formally, a filter is characterized by an impulse response $$h(t)$$, then an output signal is given by:

$y(t) = \int_{-\infty}^{\infty} h(t - \tau) x(\tau) \mathrm{d}\tau$

Or using the frequency domain, the same equation can be expressed as:

$\mathrm{Y}(j\Omega) = \mathrm{H}(j\Omega) \mathrm{X}(j\Omega)$

## Common filter classifications

#### Lowpass filter

A lowpass filter allows all low-frequency components below a specified frequency $$f_p$$, called the passband edge frequency, and blocks all high-frequency components above $$f_s$$, called the stopband edge frequency.

#### Highpass filter

A highpass filter passes all high-frequency components above $$f_p$$ and blocks all low-frequency below $$f_s$$.

#### Bandpass filter

This filter passes all frequency components between two passband edge frequencies $$f_{p1}$$ and $$f_{p2}$$ where $$f_{p1} < f_{p2}$$, and blocks all frequency components below $$f_{s1}$$ and above $$f_{s2}$$.

#### Bandstop filter

A bandstop filter blocks all frequency components between two stopband edge frequencies, $$f_{s1}$$ and $$f_{s2}$$, and allow all frequencies below and above of $$f_{p1}$$ and $$f_{p2}$$.

A bandstop filter that only blocks a specific frequency is called a notch filter.