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.

#### Addition

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*.