We first expressed a digital sequence x(n) in terms of shifted unit samples (Equation). Then the impulse response is defined. Depending on the length of its impulse response, a system is classified as finite duration impulse response (FIR) or infinite duration impulse response (IIR).

Since both the signal difference equation and impulse response are characterization of the same system, when knowing one of them we can deduce the other (section and section).

For LTI (or LSI) systems the output is the convolution of the input and the system impulse respone (Equation). This is the digital convolution summation (or sum). Remember for analog signals and systems we have the convolution integral.

The process of computing the convolution is Fold-Shift-Multiply-Add. We can use the graphical method or the method of squence (vector) to compute the convolution. When we convolve a sequence of length M with a squence of length N we will get a squence of length L=M+N−1 L=M+N−1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadYeacqGH9aqpcaWGnbGaey4kaSIaamOtaiabgkHiTiaaigdaaaa@3BE9@ (Equation).

The formula of infinite geometric series (Equation) is very useful.

The digital convolution has several useful properties, each is related to a system configuration. The commutativity (Equation and Equation) means that we can interchange the input and impulse response without affecting the output. The associativity allows the replacement of several systems in cascade by an equivalent system. The distributivity allows the replacement of several systems in paralled by an equivalent system.

When either the input signal or the system is causal , or both are causal , the limits of the convolution summation change , which we should know in order to reduce the computation.

The formula of finite geometric series (Equation) is very useful.

Stability is perhaps the most important property of real systems (hardvare or software). Stability in DSP is the bounded input - bounded output (BIBO) . The condition of stability is that the impulse response should be obsolutely summable (Equation).

Rection of systems when the applying signal is suddenly turned-on or turned-off is called transient response, which is different from the steady-state response (or stable response). Transient response is another characterization of systems. The impulse response is not related directly to the transient response. So we turn to the step response which is the reaction of the system with respect to an unit step. Another useful signal to test the transient behavior of systems is the digital rectangular pulse. Transient nature of systems are usually undesired, therefore the transient response is as short and as smooth as possible.

There are four basic frequency selective filters : Lowpass, highpass, bandpass, and bandstop (or bandsuppress) .

As for structure , we have two types of filters : Nonrecursive and recursive. In a nonrecursive filter the output only depends on the input at various instants (Equation). The filter coefficients are just the filter impulse response (Equation)). For causal filters the lower limit of the summation is zero (Equation). For most realistic cases nonrecursive filters are FIR filters. The moving average filter is a typical example , it smooths out the input signal (i.e. a lowpass filter). In real-time processing (RTP) filters must be causal .

In a recursive filter the output depends on the input at all time and on previous outputs (Equation). The filter has two types of coefficients: a k a k MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaam4Aaaqabaaaaa@37E5@ for the recursive (feedback) part, and b k b k MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadkgadaWgaaWcbaGaam4Aaaqabaaaaa@37E6@ for the nonrecursive (feedforward) part. The coefficients b k b k MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadkgadaWgaaWcbaGaam4Aaaqabaaaaa@37E6@ are not the impulse response of the filter. Recursive filters are mainly IIR. Figure is the most direct implementation of the filter. From the filter difference equation we can find the impulse response.

Concerning the types of signal processing we have on-line processing (or sample processing , or RTP) . and off-line processing (or batch processing). In off-line processing , when the sequence is

too long , we use block processing

Solution of discrete-time difference equations is similar to that of continuous-time differential equations. First, the initial conditions (or boundary conditions) must be specified. The transient response is the homogeneous root, whereas the steady-state response is the particular root of the difference equation. These two roots form the total solution.

The output signal given by the convolution of the input and impulse response is the total solution , but from this we cannot separete the homogeneous and particular.

Instead of solving the difference equation in the time domain, we can solve it in the z-transform domain.

As for convolution , correlation is defined for both analog and digital systems. Correlation of two signals measures the degree of their similarity. Correlation is used in many areas but for basic DSP convolution appears more often. The correlation between two signals is the cross-correlation (Equation) , of which a property to note is Equation. The correlation of a signal with itself is the autocorrelation (Equation), it is symmetric (Equation).

The evaluation of correlation is similar to that of convolution , except that no flipping is used.

Section gives an example of correlation in data communication . Section shows that correlation can be used to detect a periodic signal buried in random noise.