The convolution of two signals and in may be denoted `` '' and defined by
Cyclic convolution can be expressed in terms of previously defined operators as
Commutativity of Convolution
In the first step we made the change of summation variable , and in the second step, we made use of the fact that any sum over all terms is equivalent to a sum from 0 to .
Convolution as a Filtering Operation
In a convolution of two signals , where both and are signals of length (real or complex), we may interpret either or as a filter that operates on the other signal which is in turn interpreted as the filter's ``input signal''.7.5 Let denote a length signal that is interpreted as a filter. Then given any input signal , the filter output signal may be defined as the cyclic convolution of and :
As discussed below (§7.2.7), one may embed acyclic convolution within a larger cyclic convolution. In this way, real-world systems may be simulated using fast DFT convolutions (see Appendix A for more on fast convolution algorithms).
Note that only linear, time-invariant (LTI) filters can be completely represented by their impulse response (the filter output in response to an impulse at time 0). The convolution representation of LTI digital filters is fully discussed in Book II  of the music signal processing book series (in which this is Book I).
Convolution Example 1: Smoothing a Rectangular Pulse
Filter input signal .
Filter output signal .
Figure 7.3 illustrates convolution of
as graphed in Fig.7.3(c). In this case, can be viewed as a ``moving three-point average'' filter. Note how the corners of the rectangular pulse are ``smoothed'' by the three-point filter. Also note that the pulse is smeared to the ``right'' (forward in time) because the filter impulse response starts at time zero. Such a filter is said to be causal (see  for details). By shifting the impulse response left one sample to get
Convolution Example 2: ADSR Envelope
Filter output signal .
In this example, the input signal is a sequence of two rectangular pulses, creating a piecewise constant function, depicted in Fig.7.4(a). The filter impulse response, shown in Fig.7.4(b), is a truncated exponential.7.6
In this example, is again a causal smoothing-filter impulse response, and we could call it a ``moving weighted average'', in which the weighting is exponential into the past. The discontinuous steps in the input become exponential ``asymptotes'' in the output which are approached exponentially. The overall appearance of the output signal resembles what is called an attack, decay, release, and sustain envelope, or ADSR envelope for short. In a practical ADSR envelope, the time-constants for attack, decay, and release may be set independently. In this example, there is only one time constant, that of . The two constant levels in the input signal may be called the attack level and the sustain level, respectively. Thus, the envelope approaches the attack level at the attack rate (where the ``rate'' may be defined as the reciprocal of the time constant), it next approaches the sustain level at the ``decay rate'', and finally, it approaches zero at the ``release rate''. These envelope parameters are commonly used in analog synthesizers and their digital descendants, so-called virtual analog synthesizers. Such an ADSR envelope is typically used to multiply the output of a waveform oscillator such as a sawtooth or pulse-train oscillator. For more on virtual analog synthesis, see, for example, [78,77].
Convolution Example 3: Matched Filtering
Figure 7.5 illustrates convolution of
For example, could be a ``rectangularly windowed signal, zero-padded by a factor of 2,'' where the signal happened to be dc (all s). For the convolution, we need
As mentioned above, cyclic convolution can be written as
Note that when you multiply two polynomials together, their coefficients are convolved. To see this, let denote the th-order polynomial
where and are doubly infinite sequences, defined as zero for and , respectively.
Since decimal numbers are implicitly just polynomials in the powers of 10, e.g.,