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A discrete-time signal is a sequence of numbers, so-called samples. A sample occurs at Ts seconds, then another sample appears during the next Ts seconds and so on, forming the sequence. The time Ts is called sampling period and the number produced in the n-th sampling period (i.e., after n times Ts seconds) is denoted as x(nTs). For analysis purposes it is usual to assume that x(nTs) = 0 for negative values of n. In general, the samples can be complex numbers. If so, the signal is a complex signal. In these analyses, the digital frequency f is a value that expresses how much of a cycle of a sinusoidal wave is represented by a sample. In general terms, discrete-time signals may contain information for different values of f.
A digital filter is a system that receives a discrete-time signal and returns another discrete-time signal with modified characteristics (Antoniou, 2006). A useful characteristic that many digital filters exhibit is the Linear Time-Invariant (LTI) property. When the input signal x(nTs) has the value 1 for n = 0 and 0 for other values of n, the output signal of a LTI digital filter is called impulse response. LTI digital filters are often classified by the duration of its impulse response as Infinite-duration Impulse Response (IIR) or Finite-duration Impulse Response (FIR) filters (Proakis & Manolakis, 1996). Additionally, a LTI digital filter is generally described by its frequency response, which is the response of the filter to a complex exponential signal with frequency 2πf. The frequency response is a complex function of f and is periodic over f, with a period of 1/Ts. Because of this periodicity, the frequency response is just observed in the range of values from f = –1/2Ts to f = 1/2Ts. As a complex function, the frequency response has two functions associated with it: its magnitude response and its phase response.
When the values of the impulse response are real, the magnitude response is symmetric and the phase response is anti-symmetric around f = 0 (Saramaki, 1993). Thus, these responses are just observed in the range of values from f = 0 to f = 1/2Ts. Systems with real impulse response are used in a wide variety of applications, being one of the most common the traditional low-pass filter. This filter passes the frequency components of the input signal that range from 0 to fp and rejects the frequency components of the signal that range from fs to 1/2Ts, with 0 < fp < fs < 1/2Ts. The values fp and fs are the passband edge and the stopband edge frequencies, respectively, whereas the range from 0 to fp is the passband, the range from fs to 1/2Ts is the stopband and the difference fs – fp is the transition band. The ideal values of the magnitude response in passband and stopband are, respectively, 1 and 0. For a realizable filter, an acceptable deviation from these ideal values must be specified. Usually, these deviations are represented as the numbers δp and δs, respectively. Figure 1 shows the specifications of the magnitude response of a low-pass filter, along with the ideal response and the actual magnitude response of the filter, denoted as |H(f)|. For analysis purposes it is usual to assume Ts = 1, i.e., x(nTs) = x(n). Thus, in Figure 1 the range of interest of values f is from 0 to 1/2.
Figure 1. Magnitude response specifications of a low-pass filter
Key Terms in this Chapter
Complex Function: A function whose values are complex numbers.
Lineal Time-Invariant (LTI): Let us consider a filter that has as input a scaled sum of arbitrary discrete-time signals. If the output of this filter is equal to the scaled sum of the outputs of the filter to every one of these arbitrary signals, the filter is considered linear. If additionally the input signal is delayed in any possible amount and the output signal undergoes the same delay as the input, the filter is also considered time-invariant. A filter is LTI if it has both, linear and time-invariant properties.
Complex Number: A number composed by the sum of a real number a and an imaginary number i b , where b is a real number and i is defined as the square root of –1.
Multiplier: A component of a digital filter that takes the samples at its inputs and produces the multiplication of these values at its output.
Adder: A component of a digital filter that produces at its output the sum of the values of the samples present at its inputs.
Magnitude Response: A function of the frequency f where every value is obtained as the magnitude of the complex value of the frequency response in that frequency f . If the value of the frequency response in f is a complex number of the form a ( f ) + i b( f ), the magnitude of that number is given by {[ a ( f )] 2 + [ b ( f )] 2 } 1/2 .
Delay: A component of a digital filter that has one input and one output. The delay retains its input value during a sampling period T s and releases it at the next sampling interval.
Complex Exponential Signal: A signal whose samples are complex numbers, where the real and imaginary parts of the samples form, respectively, a cosine wave and a sine wave, both with the same frequency.
Phase Response: A function of the frequency f where every value is obtained as the phase of the complex value of the frequency response in that frequency f . If the value of the frequency response in f is a complex number of the form a ( f ) + i b( f ), the phase of that number is given by arctan { b ( f )/ a ( f )}.