Chirp

Generate swept-frequency cosine (chirp) signal

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Libraries:
DSP System Toolbox / Sources

Description

The Chirp block outputs a swept-frequency cosine (chirp) signal with unity amplitude and continuous phase. To specify the desired output chirp signal, you must define its instantaneous frequency function, also known as the output frequency sweep. The frequency sweep can be linear, quadratic, or logarithmic, and repeats once every Sweep time by default. For a description of the algorithms used by the Chirp block, see Algorithms.

Examples

Bidirectional Linear Sweep

In this example, the Chirp block outputs a bidirectional, linearly swept chirp signal, which is displayed by the Time Scope and the Spectrum Analyzer. You can also export the signal to the MATLAB workspace by the Signal To Workspace block.

To create a bidirectional sweep, set the Sweep mode parameter to Bidirectional. Specify the final frequency of a bidirectional sweep by setting Target time equal to Sweep time, in which case the Target frequency becomes the final frequency in the sweep. Note that in the bidirectional sweep, the period of the sweep is twice the Sweep time of the unidirectional sweep.

Run the model to see the output in the Time Scope and the Spectrum Analyzer.

You can also view the spectrogram by saving the output to the workspace and using this command:

spectrogram(dsp_examples_yout,hamming(128),110,0:.01:40,400)

Unidirectional Linear Sweep

In this example, the Chirp block outputs a unidirectional, linearly swept chirp signal. The Time Scope displays the signal output in the time domain and the Spectrum Analyzer displays the spectrogram in the frequency domain.

To obtain a unidirectional sweep with known initial and final frequency values, in the Chirp block set the Target time equal to Sweep time. In which case, the Target frequency becomes the final frequency in the sweep. Since the Target time is set to equal Sweep time (1 second), the Target frequency (25 Hz) is the final frequency of the unidirectional sweep. This technique might not work for swept cosine sweeps. For details, see the Frequency sweep described for the Frequency sweep parameter.

Run the model to see the time domain output:

When Sweep Time Is Greater than Target Time

This example shows the unexpected behavior that might arise in the Chirp block when the Sweep time is greater than Target time. The Time Scope displays the signal output, and the Spectrum Analyzer displays the spectrogram in the frequency domain.

Set the Sweep time parameter to 1.5 and specify the final frequency of a bidirectional sweep by setting Target time equal to Sweep time. The sweep reaches the Target frequency (25 Hz) at the Target time (1 second), but since Sweep time is greater than Target time, the sweep continues on its linear path until one Sweep time (1.5 seconds) is traversed.

Unexpected behavior might arise when you set Sweep time greater than Target time.

Run the model to see the chirp signal output and a spectrogram of the frequency domain.

Sweep with Negative Frequencies

In this example, the Chirp block outputs a chirp signal containing negative frequencies. The Time Scope displays the signal output in the time domain, and the Spectrum Analyzer displays the spectrogram in the frequency domain.

Set the Sweep time to 1.5, Initial frequency to 25, Target frequency to 0, and Target time equal to Sweep time. The output chirp of this example might not behave as you expect because the sweep contains negative frequencies between 1 and 1.5 seconds. The sweep reaches the Target frequency (0 Hz) at 1 second, then continues on its negative slope, taking on negative frequency values until it traverses one Sweep time (1.5 seconds).

Run the model to see the time domain output.

Aliased Sweep

In this example, the Chirp block outputs a chirp signal containing negative frequencies. The Time Scope displays the signal output in the time domain, and the Spectrum Analyzer displays the spectrogram in the frequency domain.

Set the Target frequency to 275 and specify Target time equal to Sweep time. The output chirp of this example might not behave as you expect because the sweep contains frequencies greater than half the sampling frequency (200 Hz). If you unexpectedly get a chirp output with a spectrogram resembling the one following, your chirp's sweep might contain frequencies greater than half the sampling frequency.

Run the model to see the signal output and the spectrogram.

Extended Examples

Streaming Power Spectrum Estimation Using Welch's Method

Use Welch's method of averaged modified periodogram to estimate power spectrum.

• Open Script

Ports

Output

Port_1 — Swept-frequency cosine (chirp) signal

scalar | vector

Swept-frequency cosine (chirp) signal. In Linear, Logarithmic, and Quadratic modes (set by the Frequency sweep parameter), the block outputs a swept-frequency cosine with instantaneous frequency values specified by the frequency and time parameters. In Swept cosine mode, the block outputs a swept-frequency cosine with a linear instantaneous output frequency that may differ from the one specified by the frequency and time parameters.

For more information about how the block computes the output, see Algorithms.

Data Types: single | double

Parameters

Frequency sweep — Type of frequency sweep

Linear (default) | Swept cosine | Logarithmic | Quadratic

The type of output instantaneous frequency sweep, f_i(t): Linear, Logarithmic, Quadratic, or Swept cosine. For more information, see Shaping the Frequency Sweep and Algorithms.

Limitations

When you want a linearly swept chirp signal, we recommend that you use a Linear frequency sweep. Though a Swept cosine frequency sweep also yields a linearly swept chirp signal, the output might have unexpected frequency content.

• The swept cosine sweep value at the Target time is not necessarily the Target frequency. This is because the user-specified sweep is not the actual frequency sweep of the swept cosine output, as noted in Output Computation Method for Swept Cosine Frequency Sweep. See the table Instantaneous Frequency Sweep Values for the actual value of the swept cosine sweep at the Target time.

• In Swept cosine mode, do not set the parameters so that 1/T_sw is much greater than the values of the Initial frequency and Target frequency parameters. In such cases, the actual frequency content of the swept cosine sweep might be closer to 1/T_sw, far exceeding the Initial frequency and Target frequency parameter values.

Sweep mode — Sweep mode

Unidirectional (default) | Bidirectional

The Sweep mode parameter determines whether your sweep is unidirectional or bidirectional, which affects the shape of your output frequency sweep (see Shaping the Frequency Sweep). The following table describes the characteristics of unidirectional and bidirectional sweeps.

Sweep Mode Parameter Settings	Sweep Characteristics
Unidirectional	Lasts for one Sweep time, Tsw Repeats once every Tsw
Bidirectional	Lasts for twice the Sweep time, 2Tsw Repeats once every 2Tsw First half is identical to its unidirectional counterpart. Second half is a mirror image of the first half.

The following diagram illustrates a linear sweep in both sweep modes. For information on setting the frequency values in your sweep, see Setting Instantaneous Frequency Sweep Values.

Initial frequency (Hz) — Initial frequency

1000 (default) | scalar

For Linear, Quadratic, and Swept cosine sweeps, the initial frequency, f₀, of the output chirp signal. You can specify the Initial frequency (Hz) as a scalar, greater than or equal to zero. For Logarithmic sweeps, Initial frequency is one less than the actual initial frequency of the sweep. Also, when the sweep is Logarithmic, you must set the Initial frequency to be less than the Target frequency.

For more information, see Setting Instantaneous Frequency Sweep Values.

Tunable: Yes

Target frequency (Hz) — Target frequency

4000 (default) | scalar

For Linear, Quadratic, and Logarithmic sweeps, the instantaneous frequency, f_i(t_g), of the output at the Target time, t_g. You can specify the Target frequency (Hz) as a scalar, greater than or equal to zero. For a Swept cosine sweep, Target frequency is the instantaneous frequency of the output at half the Target time, t_g/2. When Frequency sweep is Logarithmic, you must set the Target frequency to be greater than the Initial frequency.

For more information, see Setting Instantaneous Frequency Sweep Values.

Tunable: Yes

Target time (s) — Target time of sweep

1 (default) | scalar

For Linear, Quadratic, and Logarithmic sweeps, the time, t_g, at which the sweep reaches the Target frequency, f_i(t_g). For a Swept cosine sweep, Target time is the time at which the sweep reaches 2f_i(t_g) - f₀. Target time must be a scalar that is greater than or equal to zero, and less than or equal to Sweep time , $$T_{sw}\ge t_g$$.

For more information, see Setting Instantaneous Frequency Sweep Values.

Tunable: Yes

Sweep time (s) — Sweep time

1 (default) | scalar

In Unidirectional Sweep mode, the Sweep time, T_sw, is the period of the output frequency sweep. In Bidirectional Sweep mode, the Sweep time is half the period of the output frequency sweep. Sweep time must be a scalar that is greater than or equal to Target time, $$T_{sw}\ge t_g$$.

Tunable: Yes

Initial phase (rad) — Initial phase of cosine output

0 (default) | scalar

The phase, $${\varphi }_0$$, of the cosine output at t=0; $$y_{chirp}(t)=\cos ({\varphi }_0)$$. You can specify the Initial phase (rad) as a scalar that is greater than or equal to zero.

Tunable: Yes

Sample time — Output sample period

1/8000 (default) | positive scalar

The sample period, T_s, of the output. The output frame period is M_oT_s, where M_o is the number of samples per frame.

Samples per frame — Samples per frame

1 (default) | positive integer

The number of samples, M_o, to buffer into each output frame, specified as a positive integer scalar.

Output data type — Output data type

Double (default) | Single

The data type of the output, specified as single precision or double precision.

Block Characteristics

Data Types	double | single
Direct Feedthrough	no
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no

More About

Shaping the Frequency Sweep

You control the basic shape of the output instantaneous frequency sweep, f_i(t), using the Frequency sweep and Sweep mode parameters.

Parameters for Setting Sweep Shape	Possible Setting	Parameter Description
Frequency sweep	Linear Quadratic Logarithmic Swept cosine	Determines whether the sweep frequencies vary linearly, quadratically, or logarithmically. Linear and swept cosine sweeps both vary linearly.
Sweep mode	Unidirectional Bidirectional	Determines whether the sweep is unidirectional or bidirectional. For details, see Sweep mode

The following diagram illustrates the possible shapes of the frequency sweep that you can obtain by setting the Frequency sweep and Sweep mode parameters.

For information on how to set the frequency values in your sweep, see Setting Instantaneous Frequency Sweep Values.

Setting Instantaneous Frequency Sweep Values

Set the following parameters to tune the frequency values of your output frequency sweep.

• Initial frequency (Hz), f₀

• Target frequency (Hz), f_i(t_g)

• Target time (seconds), t_g

The following table summarizes the sweep values at specific times for all Frequency sweep settings. For information on the formulas used to compute sweep values at other times, see Algorithms.

Instantaneous Frequency Sweep Values

Frequency Sweep	Sweep Value at t = 0	Sweep Value at t = t g	Time When Sweep Value Is Target Frequency, f i ( t g )
Linear	f0	fi(tg)	tg
Quadratic	f0	fi(tg)	tg
Logarithmic	f0	fi(tg)	tg
Swept cosine	f0	2fi(tg) - f0	tg/2

Algorithms

The Chirp block uses one of two formulas to compute the block output, depending on the Frequency Sweep parameter setting. For details, see the following sections.

Equations for Output Computation

The following table shows the equations used by the block to compute the user-specified output frequency sweep, f_i(t), the block output, y_chirp(t), and the actual output frequency sweep, f_i(actual)(t). The only time the user-specified sweep is not the actual output sweep is when the Frequency sweep parameter is set to Swept cosine.

Note

The following equations apply only to unidirectional sweeps in which f_i(0) < f_i(t_g). To derive equations for other cases, examine the following table and the diagram in Shaping the Frequency Sweep.

The table of equations used by the block contains the following variables:

• f_i(t) — the user-specified frequency sweep

• f_i(actual)(t) — the actual output frequency sweep, usually equal to f_i(t)

• y(t) — the Chirp block output

• $$\psi (t)$$ — the phase of the chirp signal, where $$\psi (0)=0$$, and $$2\pi f_i(t)$$ is the derivative of the phase

  $$f_i(t)=\frac{1}{2\pi }\cdot \frac{d\psi (t)}{dt}$$

• $${\varphi }_0$$ — the Initial phase parameter value, where $$y_{chirp}(0)=\cos ({\varphi }_0)$$

Equations for Unidirectional Positive Sweeps

Frequency Sweep	Block Output Chirp Signal	User-Specified Frequency Sweep, f i ( t )	$$\beta$$	Actual Frequency Sweep, f i(actual) ( t )
Linear	$$y(t)=\cos (\psi (t)+{\varphi }_0)$$	$$f_i(t)=f_0+\beta t$$	$$\beta =\frac{f_i(t_g)-f_0}{t_g}$$	$$f_i{{}_{(actual)}}^{(t)}=f_i{}^{(t)}$$
Quadratic	Same as Linear	$$f_i(t)=f_0+\beta t^2$$	$$\beta =\frac{f_i(t_g)-f_0}{t_g^2}$$	$$f_i{{}_{(actual)}}^{(t)}=f_i{}^{(t)}$$
Logarithmic	Same as Linear	$$F_i(t)=f_0{\left(\frac{f_i\left(t_g\right)}{f_0}\right)}^{\frac{t}{t_g}}$$ Where fi(tg) > f0> 0	N/A	$$f_{i(actual)}(t)=f_i(t)$$
Swept cosine	$$y(t)=\cos (2\pi f_i(t)t+{\varphi }_0)$$	Same as Linear	Same as Linear	$$f_{i(actual)}(t)=f_i(t)+\beta t$$

Output Computation Method for Linear, Quadratic, and Logarithmic Frequency Sweeps

The derivative of the phase of a chirp function gives the instantaneous frequency of the chirp function. The Chirp block uses this principle to calculate the chirp output when the Frequency Sweep parameter is set to Linear, Quadratic, or Logarithmic.

$$y_{chirp}(t)=\cos (\psi (t)+{\varphi }_0)$$	Linear, quadratic, or logarithmic chirp signal with phase $$\psi (t)$$
$$f_i(t)=\frac{1}{2\pi }\cdot \frac{d\psi (t)}{dt}$$	Phase derivative is instantaneous frequency

For instance, if you want a chirp signal with a linear instantaneous frequency sweep, set the Frequency Sweep parameter to Linear, and tune the linear sweep values by setting other parameters appropriately. The block outputs a chirp signal, the phase derivative of which is the specified linear sweep. This ensures that the instantaneous frequency of the output is the linear sweep you desired. For equations describing the linear, quadratic, and logarithmic sweeps, see Equations for Output Computation.

Output Computation Method for Swept Cosine Frequency Sweep

To generate the swept cosine chirp signal, the block sets the swept cosine chirp output as follows.

$$y_{chirp}(t)=\cos (\psi (t)+{\varphi }_0)=\cos (2\pi f_i(t)t+{\varphi }_0)$$

Swept cosine chirp output (Instantaneous frequency equation, does not hold.)

The instantaneous frequency equation, shown in Output Computation Method for Linear, Quadratic, and Logarithmic Frequency Sweeps, does not hold for the swept cosine chirp, so the user-defined frequency sweep, f_i(t), is not the actual output frequency sweep, f_i(actual)(t), of the swept cosine chirp. Thus, the swept cosine output might not behave as you expect. To learn more about swept cosine chirp behavior, see Frequency sweep described for the Frequency sweep parameter and Equations for Output Computation.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced before R2006a