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.