# freqs

Frequency response of analog filters

## Syntax

``h = freqs(b,a,w)``
``[h,wout] = freqs(b,a,n)``
``freqs(___)``

## Description

example

````h = freqs(b,a,w)` returns the complex frequency response of the analog filter specified by the coefficient vectors `b` and `a`, evaluated at the angular frequencies `w`.```

example

````[h,wout] = freqs(b,a,n)` uses `n` frequency points to compute `h` and returns the corresponding angular frequencies in `wout`.```

example

````freqs(___)` with no output arguments plots the magnitude and phase responses as functions of angular frequency in the current figure window. You can use this syntax with either of the previous input syntaxes.```

## Examples

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Find and graph the frequency response of the transfer function

`$H\left(s\right)=\frac{0.2{s}^{2}+0.3s+1}{{s}^{2}+0.4s+1}.$`

```a = [1 0.4 1]; b = [0.2 0.3 1]; w = logspace(-1,1); h = freqs(b,a,w); mag = abs(h); phase = angle(h); phasedeg = phase*180/pi; subplot(2,1,1) loglog(w,mag) grid on xlabel('Frequency (rad/s)') ylabel('Magnitude') subplot(2,1,2) semilogx(w,phasedeg) grid on xlabel('Frequency (rad/s)') ylabel('Phase (degrees)')```

You can also generate the plots by calling `freqs` with no output arguments.

```figure freqs(b,a,w)```

Design a 5th-order analog Butterworth lowpass filter with a cutoff frequency of 2 GHz. Multiply by $2\pi$ to convert the frequency to radians per second. Compute the frequency response of the filter at 4096 points.

```n = 5; fc = 2e9; [zb,pb,kb] = butter(n,2*pi*fc,"s"); [bb,ab] = zp2tf(zb,pb,kb); [hb,wb] = freqs(bb,ab,4096);```

Design a 5th-order Chebyshev Type I filter with the same edge frequency and 3 dB of passband ripple. Compute its frequency response.

```[z1,p1,k1] = cheby1(n,3,2*pi*fc,"s"); [b1,a1] = zp2tf(z1,p1,k1); [h1,w1] = freqs(b1,a1,4096);```

Design a 5th-order Chebyshev Type II filter with the same edge frequency and 30 dB of stopband attenuation. Compute its frequency response.

```[z2,p2,k2] = cheby2(n,30,2*pi*fc,"s"); [b2,a2] = zp2tf(z2,p2,k2); [h2,w2] = freqs(b2,a2,4096);```

Design a 5th-order elliptic filter with the same edge frequency, 3 dB of passband ripple, and 30 dB of stopband attenuation. Compute its frequency response.

```[ze,pe,ke] = ellip(n,3,30,2*pi*fc,"s"); [be,ae] = zp2tf(ze,pe,ke); [he,we] = freqs(be,ae,4096);```

Design a 5th-order Bessel filter with the same edge frequency. Compute its frequency response.

```[zf,pf,kf] = besself(n,2*pi*fc); [bf,af] = zp2tf(zf,pf,kf); [hf,wf] = freqs(bf,af,4096);```

Plot the attenuation in decibels. Express the frequency in gigahertz. Compare the filters.

```plot([wb w1 w2 we wf]/(2e9*pi), ... mag2db(abs([hb h1 h2 he hf]))) axis([0 5 -45 5]) grid xlabel("Frequency (GHz)") ylabel("Attenuation (dB)") legend(["butter" "cheby1" "cheby2" "ellip" "besself"])```

The Butterworth and Chebyshev Type II filters have flat passbands and wide transition bands. The Chebyshev Type I and elliptic filters roll off faster but have passband ripple. The frequency input to the Chebyshev Type II design function sets the beginning of the stopband rather than the end of the passband. The Bessel filter has approximately constant group delay along the passband.

Design a 5th-order analog lowpass Bessel filter with an approximately constant group delay up to $1{0}^{4}$ rad/s. Plot the frequency response of the filter using `freqs`.

```[b,a] = besself(5,10000); % Bessel analog filter design freqs(b,a) % Plot frequency response```

## Input Arguments

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Transfer function coefficients, specified as vectors.

Example: `[b,a] = butter(5,50,'s')` specifies a fifth-order Butterworth filter with a cutoff frequency of 50 rad/second.

Data Types: `single` | `double`

Angular frequencies, specified as a positive real vector expressed in rad/second.

Example: `2*pi*logspace(6,9)` specifies 50 logarithmically spaced angular frequencies from 1 MHz (2π × 106 rad/second) and 1 GHz (2π × 109 rad/second).

Data Types: `single` | `double`

Number of evaluation points, specified as a positive integer scalar.

Data Types: `single` | `double`

## Output Arguments

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Frequency response, returned as a vector.

Angular frequencies at which `h` is computed, returned as a vector.

## Algorithms

`freqs` returns the complex frequency response of an analog filter specified by `b` and `a`. The function evaluates the ratio of Laplace transform polynomials

`$H\left(s\right)=\frac{B\left(s\right)}{A\left(s\right)}=\frac{b\left(1\right){s}^{n}+b\left(2\right){s}^{n-1}+\cdots +b\left(n+1\right)}{a\left(1\right){s}^{m}+a\left(2\right){s}^{m-1}+\cdots +a\left(m+1\right)}$`

along the imaginary axis at the frequency points s = :

```s = 1j*w; h = polyval(b,s)./polyval(a,s); ```

## Version History

Introduced before R2006a