# Frequency Response Analysis

This chapter discusses*frequency-response analysis*of digital filters. The frequency response is a complex function which yields the gain and phase-shift as a function of frequency. Useful variants such as

*phase delay*and

*group delay*are defined, and examples and applications are considered.

## Frequency Response

The*frequency response*of an LTI filter may be defined as the spectrum of the output signal divided by the spectrum of the input signal. In this section, we show that the frequency response of any LTI filter is given by its transfer function evaluated on the unit circle,

*i.e.*, . We then show that this is the same result we got using sine-wave analysis in Chapter 1. Beginning with Eq.(6.4), we have

*z*transform of the filter input signal , is the

*z*transform of the output signal , and is the filter transfer function. A basic property of the

*z*transform is that, over the unit circle , we find the

*spectrum*[84].

^{8.1}To show this, we set in the definition of the

*z*transform, Eq.(6.1), to obtain

*bilateral*

*discrete time Fourier transform (DTFT)*when is normalized to 1 [59,84]. When is causal, this definition reduces to the usual (unilateral) DTFT definition:

Applying this relation to gives

Thus, the spectrum of the filter output is just the input spectrum times the spectrum of the impulse response . We have therefore shown the following:

This immediately implies the following:

We can express this mathematically by writing

*gain*and

*phase shift*applied by the filter at each frequency. Since , , and are constants, the frequency response is only a function of radian frequency . Since is real, the frequency response may be considered a

*complex-valued function of a real variable*. The response at frequency Hz, for example, is , where is the sampling period in seconds. It might be more convenient to define new functions such as and write simply instead of having to write so often, but doing so would add a lot of new functions to an already notation-rich scenario. Furthermore, writing makes explicit the connection between the transfer function and the frequency response. Notice that defining the frequency response as a function of places the frequency ``axis'' on the

*unit circle*in the complex plane, since . As a result, adding multiples of the sampling frequency to corresponds to traversing whole cycles around the unit circle, since

*analytic continuation*(§D.2). Since analytic continuation is unique (for all filters encountered in practice), we get the same result going either direction. Because every complex number can be represented as a magnitude and angle ,

*viz.*, , the frequency response may be decomposed into two real-valued functions, the

*amplitude response*and the

*phase response*. Formally, we may define them as follows:

## Amplitude Response

Definition.Theamplitude responseof an LTI filter is defined as themagnitude(or modulus) of the (complex) filter frequency response ,i.e.,

*magnitude frequency response*. The real-valued amplitude response specifies the amplitude

*gain*that the filter provides at each frequency .

## Phase Response

Definition.Thephase responseof an LTI filter is defined as the phase (or angle) of the frequency response :

*filter phase*can be defined unambiguously as the phase of its frequency response. The real-valued phase response gives the

*phase shift*in radians that each input component sinusoid will undergo.

## Polar Form of the Frequency Response

When the complex-valued frequency response is expressed in*polar form*, the amplitude response and phase response explicitly appear:

Writing the basic frequency response description

*times*the filter amplitude response, and the output phase equals the input phase

*plus*the filter phase at each frequency . Equation (7.3) gives the frequency response in polar form. For completeness, recall the transformations between polar and rectangular forms (

*i.e.*, for converting real and imaginary parts to magnitude and angle, and vice versa):

#### Separating the Transfer Function Numerator and Denominator

From Eq.(6.5) we have that the transfer function of a recursive filter is a ratio of polynomials in :where

## Frequency Response as a Ratio of DTFTs

From Eq.(6.5), we have , so that the frequency response isThis expression provides a convenient basis for the computation of an IIR frequency response in software, as we pursue further in the next section.

### Frequency Response in Matlab

In practice, we usually work with a*sampled*frequency axis. That is, instead of evaluating the transfer function at to obtain the frequency response , where is

*continuous*radian frequency, we compute instead

*sampled DTFT*of divided by the

*sampled DTFT*of :

*discrete Fourier transform (DFT)*[84]. Thus, we can write

DFT

and
, where denotes the
sampling rate in Hz.
To avoid undersampling
, we must have , and to
avoid undersampling
, we must have . In general,
*will be undersampled*(when ), because it is the quotient of over . This means, for example, that computing the impulse response from the sampled frequency response will be

*time aliased*in general.

*I.e.*,

IDFT

will be time-aliased in the IIR case. In other words, an infinitely
long impulse response cannot be Fourier transformed using a
finite-length DFT, and this corresponds to not being able to sample
the frequency response of an IIR filter without some loss of
information. In practice, we simply choose sufficiently large
so that the sampled frequency response is accurate enough for our
needs. A conservative practical rule of thumb when analyzing stable
digital filters is to choose
, where
denotes the maximum pole magnitude. This choice
provides more than 60 dB of decay in the impulse response over a
duration of samples, which is the time-aliasing block size.
(The time to decay 60 dB, or ``'', is a little less than
time constants [84], and the time-constant of decay for a
single pole at radius can be approximated by samples,
when is close to 1, as derived in §8.6.)
As is well known, when the DFT length is a power of 2, *e.g.*, , the DFT can be computed extremely efficiently using the

*Fast Fourier Transform (FFT)*. Figure 7.1 gives an example matlab script for computing the frequency response of an IIR digital filter using two FFTs. The Matlab function

`freqz`also uses this method when possible (

*e.g.*, when is a power of 2).

function [H,w] = myfreqz(B,A,N,whole,fs) %MYFREQZ Frequency response of IIR filter B(z)/A(z). % N = number of uniform frequency-samples desired % H = returned frequency-response samples (length N) % w = frequency axis for H (length N) in radians/sec % Compatible with simple usages of FREQZ in Matlab. % FREQZ(B,A,N,whole) uses N points around the whole % unit circle, where 'whole' is any nonzero value. % If whole=0, points go from theta=0 to pi*(N-1)/N. % FREQZ(B,A,N,whole,fs) sets the assumed sampling % rate to fs Hz instead of the default value of 1. % If there are no output arguments, the amplitude and % phase responses are displayed. Poles cannot be % on the unit circle. A = A(:).'; na = length(A); % normalize to row vectors B = B(:).'; nb = length(B); if nargin < 3, N = 1024; end if nargin < 4, if isreal(b) & isreal(a), whole=0; else whole=1; end; end if nargin < 5, fs = 1; end Nf = 2*N; if whole, Nf = N; end w = (2*pi*fs*(0:Nf-1)/Nf)'; H = fft([B zeros(1,Nf-nb)]) ./ fft([A zeros(1,Nf-na)]); if whole==0, w = w(1:N); H = H(1:N); end if nargout==0 % Display frequency response if fs==1, flab = 'Frequency (cyles/sample)'; else, flab = 'Frequency (Hz)'; end subplot(2,1,1); % In octave, labels go before plot: plot([0:N-1]*fs/N,20*log10(abs(H)),'-k'); grid('on'); xlabel(flab'); ylabel('Magnitude (dB)'); subplot(2,1,2); plot([0:N-1]*fs/N,angle(H),'-k'); grid('on'); xlabel(flab); ylabel('Phase'); end |

###
Example LPF Frequency Response Using `freqz`

Figure 7.2 lists a short matlab program illustrating usage of
`freqz`in Octave (as found in the

`octave-forge`package). The same code should also run in Matlab, provided the Signal Processing Toolbox is available. The lines of code not pertaining to plots are the following:

[B,A] = ellip(4,1,20,0.5); % Design lowpass filter B(z)/A(z) [H,w] = freqz(B,A); % Compute frequency response H(w)The filter example is a recursive fourth-order elliptic function lowpass filter cutting off at half the Nyquist limit (``'' in the fourth argument to

`ellip`). The maximum passband ripple

^{8.2}is 1 dB (2nd argument), and the maximum stopband ripple is 20 dB (3rd arg). The sampled frequency response is returned in the

`H`array, and the specific radian frequency samples corresponding to

`H`are returned in the

`w`(``omega'') array. An immediate plot can be obtained in either Matlab or Octave by simply typing

plot(w,abs(H)); plot(w,angle(H));However, the example of Fig.7.2 uses more detailed ``compatibility'' functions listed in Appendix J. In particular, the

`freqplot`utility is a simple compatibility wrapper for

`plot`with label and title support (see §J.2 for Octave and Matlab version listings), and

`saveplot`is a trivial compatibility wrapper for the

`freqplot`plots are shown in Fig.7.3(a) and Fig.7.3(b).

^{8.3}

[B,A] = ellip(4,1,20,0.5); % Design the lowpass filter [H,w] = freqz(B,A); % Compute its frequency response % Plot the frequency response H(w): % figure(1); freqplot(w,abs(H),'-k','Amplitude Response',... 'Frequency (rad/sample)', 'Gain'); saveplot('../eps/freqzdemo1.eps'); figure(2); freqplot(w,angle(H),'-k','Phase Response',... 'Frequency (rad/sample)', 'Phase (rad)'); saveplot('../eps/freqzdemo2.eps'); % Plot frequency response in a "multiplot" like Matlab uses: % figure(3); plotfr(H,w/(2*pi)); if exist('OCTAVE_VERSION') disp('Cannot save multiplots to disk in Octave') else saveplot('../eps/freqzdemo3.eps'); end |

Amplitude
Response
Phase Response |

## Phase and Group Delay

In the previous sections we looked at the two most important frequency-domain representations for LTI digital filters, the transfer function and the frequency response:*phase delay*and

*group delay*. After considering some examples and special cases,

*poles*and

*zeros*of the transfer function are discussed in the next chapter.

### Phase Delay

The phase response of an LTI filter gives the radian phase shift added to the phase of each sinusoidal component of the input signal. It is often more intuitive to consider instead the*phase delay*, defined as

*time delay*in seconds experienced by each sinusoidal component of the input signal. For example, in the simplest lowpass filter of Chapter 1, we found that the phase response was , which corresponds to a phase delay , or one-half sample. Thus, we can say precisely that the filter exhibits half a sample of time delay at every frequency. (Regarding the discussion in §1.3.2, it is now obvious how we should define the filter phase response at frequencies 0 and .) From a sinewave-analysis point of view, if the input to a filter with frequency response is

### Phase Unwrapping

In working with phase delay, it is often necessary to ``unwrap'' the phase response . Phase unwrapping ensures that all appropriate multiples of have been included in . We defined simply as the complex angle of the frequency response , and this is not sufficient for obtaining a phase response which can be converted to true time delay. If multiples of are discarded, as is done in the definition of complex angle, the phase delay is modified by multiples of the sinusoidal period. Since LTI filter analysis is based on sinusoids without beginning or end, one cannot in principle distinguish between ``true'' phase delay and a phase delay with discarded sinusoidal periods when looking at a sinusoidal output at any given frequency. Nevertheless, it is often useful to define the filter phase response as a*continuous*function of frequency with the property that or (for real filters). This specifies how to

*unwrap*the phase response at all frequencies where the amplitude response is finite and nonzero. When the amplitude response goes to zero or infinity at some frequency, we can try to take a limit from below and above that frequency. Matlab and Octave have a function called

`unwrap()`

which
implements a numerical algorithm for phase unwrapping.
Figures 7.6.2 and 7.6.2 show the effect of the
`unwrap`function on the phase response of the example elliptic lowpass filter of §7.5.2, modified to contract the zeros from the unit circle to a circle of radius in the plane:

[B,A] = ellip(4,1,20,0.5); % design lowpass filter B = B .* (0.95).^[1:length(B)]; % contract zeros by 0.95 [H,w] = freqz(B,A); % frequency response theta = angle(H); % phase response thetauw = unwrap(theta); % unwrapped phase responseIn Fig.7.6.2, the phase-response minimum has ``wrapped around'' to the top of the plot. In Fig.7.6.2, the phase response is continuous. We have contracted the zeros away from the unit circle in this example, because the phase response really does switch discontinuously by radians when frequency passes through a point where the phases crosses zero along the unit circle (see Fig.7.3(b)). The

`unwrap`function need not modify these discontinuities, but it is free to add or subtract any integer multiple of in order to obtain the ``best looking'' discontinuity. Typically, for best results, such discontinuities should

*alternate*between and , making the phase response resemble a distorted ``square wave'', as in Fig.7.3(b). A more precise example appears in Fig.10.2.

Phase
Response
Unwrapped Response |

### Group Delay

A more commonly encountered representation of filter phase response is called the*group delay*, defined by

*i.e.*, for some constant , the group delay and the phase delay are identical, and each may be interpreted as time delay (equal to samples when ). If the phase response is nonlinear, then the relative phases of the sinusoidal signal components are generally altered by the filter. A nonlinear phase response normally causes a ``smearing'' of attack transients such as in percussive sounds. Another term for this type of phase distortion is

*phase dispersion*. This can be seen below in §7.6.5. An example of a linear phase response is that of the simplest lowpass filter, . Thus, both the phase delay and the group delay of the simplest lowpass filter are equal to half a sample at every frequency. For any reasonably smooth phase function, the group delay may be interpreted as the

*time delay of the amplitude envelope*of a sinusoid at frequency [63]. The bandwidth of the amplitude envelope in this interpretation must be restricted to a frequency interval over which the phase response is approximately linear. We derive this result in the next subsection. Thus, the name ``group delay'' for refers to the fact that it specifies the delay experienced by a narrow-band ``group'' of sinusoidal components which have frequencies within a narrow frequency interval about . The width of this interval is limited to that over which is approximately constant.

#### Derivation of Group Delay as Modulation Delay

Suppose we write a narrowband signal centered at frequency aswhere is defined as the

*carrier frequency*(in radians per sample), and is some ``lowpass''

*amplitude modulation*signal. The modulation can be complex-valued to represent either phase or amplitude modulation or both. By ``lowpass,'' we mean that the spectrum of is concentrated near dc,

*i.e.*,

Assuming the phase response is approximately linear over the narrow frequency interval , we can write

*i.e.*, using a Fourier superposition integral for ) gives

### Group Delay Examples in Matlab

Figure 7.6 compares the group delay responses for a number of classic lowpass filters, including the example of Fig.7.2. The matlab code is listed in Fig.7.5. See,*e.g.*, Parks and Burrus [64] for a discussion of Butterworth, Chebyshev, and Elliptic Function digital filter design. See also §I.2 for details on the Butterworth case. The various types may be summarized as follows:

- Butterworth filters are maximally flat in middle of the passband.
- Chebyshev Type I filters are ``equiripple'' in the passband and ``Butterworth'' in the stopband.
- Chebyshev Type II filters are ``Butterworth'' in the passband and equiripple in the stopband.
- Elliptic function filters are equiripple in both the passband and stopband.

*equiripple*'' means ``equal ripple''; that is, the error oscillates with equal peak magnitudes across the band. An equiripple error characterizes

*optimality in the Chebyshev sense*[64,78]. As Fig.7.6.4 indicates, and as is well known, the Butterworth filter has the flattest group delay curve (and most gentle transition from passband to stopband) for the four types compared. The elliptic function filter has the largest amount of ``delay distortion'' near the cut-off frequency (passband edge frequency). Fundamentally, the more abrupt the transition from passband to stopband, the greater the delay-distortion across that transition, for any minimum-phase filter. (Minimum-phase filters are introduced in Chapter 11.) The delay-distortion can be compensated by

*delay equalization*,

*i.e.*, adding delay at other frequencies in order approach an overall constant group delay versus frequency. Delay equalization is typically carried out using an allpass filter (defined in §B.2) in series with the filter to be delay-equalized [1].

[Bb,Ab] = butter(4,0.5); % order 4, cutoff at 0.5 * pi Hb=freqz(Bb,Ab); Db=grpdelay(Bb,Ab); [Bc1,Ac1] = cheby1(4,1,0.5); % 1 dB passband ripple Hc1=freqz(Bc1,Ac1); Dc1=grpdelay(Bc1,Ac1); [Bc2,Ac2] = cheby2(4,20,0.5); % 20 dB stopband attenuation Hc2=freqz(Bc2,Ac2); Dc2=grpdelay(Bc2,Ac2); [Be,Ae] = ellip(4,1,20,0.5); % like cheby1 + cheby2 He=freqz(Be,Ae); [De,w]=grpdelay(Be,Ae); figure(1); plot(w,abs([Hb,Hc1,Hc2,He])); grid('on'); xlabel('Frequency (rad/sample)'); ylabel('Gain'); legend('butter','cheby1','cheby2','ellip'); saveplot('../eps/grpdelaydemo1.eps'); figure(2); plot(w,[Db,Dc1,Dc2,De]); grid('on'); xlabel('Frequency (rad/sample)'); ylabel('Delay (samples)'); legend('butter','cheby1','cheby2','ellip'); saveplot('../eps/grpdelaydemo2.eps'); |

Group Delays |

### Vocoder Analysis

The definitions of phase delay and group delay apply quite naturally to the analysis of the*vocoder*(``voice coder'') [21,26,54,76]. The vocoder provides a bank of bandpass filters which decompose the input signal into narrow spectral ``slices.'' This is the analysis step. For synthesis (often called

*additive synthesis*), a bank of sinusoidal oscillators is provided, having amplitude and frequency control inputs. The oscillator frequencies are tuned to the filter center frequencies, and the amplitude controls are driven by the amplitude envelopes measured in the filter-bank analysis. (Typically, some data reduction or envelope modification has taken place in the amplitude envelope set.) With these oscillators, the band slices are independently regenerated and summed together to resynthesize the signal. Suppose we excite only channel of the vocoder with the input signal

*amplitude modulated sinusoid*. The component can be called the

*carrier wave*, while is the

*amplitude envelope*. If the phase of each channel filter is linear in frequency within the passband (or at least across the width of the spectrum of ), and if each channel filter has a flat amplitude response in its passband, then the filter output will be, by the analysis of the previous section,

where is the phase delay of the channel filter at frequency , and is the group delay at that frequency. Thus, in vocoder analysis for additive synthesis, the phase delay of the analysis filter bank gives the time delay experienced by the oscillator carrier waves, while the group delay of the analysis filter bank gives the time delay imposed on the estimated oscillator amplitude-envelope functions. Note that a nonlinear phase response generally results in , and for . As a result, the

*dispersive*nature of additive synthesis reconstruction in this case can be seen in Eq.(7.8).

### Numerical Computation of Group Delay

The definition of group delay,*logarithmic derivative*of the frequency response. Expressing the frequency response in polar form as

imim

Consider first the FIR case in which , with
In this case, the derivative is simply

*i.e.*, the th coefficient of the polynomial is , for . In matlab, we may compute

`Br`from

`B`via the following statement:

Br = B .* [0:M]; % Compute ramped B polynomialThe group delay of an FIR filter can now be written as

imimre

In matlab, the group delay, in samples, can be computed simply asD = real(fft(Br) ./ fft(B))where the

`fft`, of course, approximates the Discrete Time Fourier Transform (DTFT). Such sampling of the frequency axis by this approximation is information-preserving whenever the number of samples (FFT length) exceeds the polynomial order . The

*ratio*of sampled DTFTs, however, is

*undersampled*, in general. In fact, we may have at some frequencies (``zeros on the unit circle''). The

`grpdelay`matlab utility in §J.8 watches out for division by zero, and simply sets the group delay to zero at such frequencies. Note that the true group delay approaches infinite magnitude as either a zero or pole approaches the unit circle. Finally, when there are both poles and zeros, we have

Straightforward differentiation yields

and this can be implemented analogous to the FIR case discussed above. However, a faster algorithm (usually) results from converting the IIR case to the FIR case:

where

^{8.4}In matlab, the C polynomial is given by

C = conv(B,fliplr(conj(A)));It is straightforward to show (Problem 11) that

re

This method is implemented in §J.8.
## Frequency Response Analysis Problems

See`http://ccrma.stanford.edu/~jos/filtersp/Frequency_Response_Analysis_Problems.html`

**Next Section:**

Pole-Zero Analysis

**Previous Section:**

Transfer Function Analysis