Lab #9: Everyday Sinusoidal Signals
Lab #9 description is based on Lab A of Lab C.7 in Appendix C of the text, but the
warm-up has been changed quite a bit.
Formal Lab Report: You must write a formal lab report that describes your system
for DTMF decoding (Section 4). You should read the Pre-Lab section of the lab
and do all the exercises in the Pre-Lab section before your assigned lab time.
Important: When it instructs you to get an updated matlab file (like
The lab introduces a practical application where sinusoidal signals are used to
transmit information: a touchtone dialer. Bandpass FIR filters can be used to
extract the information encoded in the waveforms. The goal of this lab is to deaign
and implement bandpass FIR filters in MATLAB, and do the decoding automatically.
In the experiements of this lab, you will use firfilt(), or conv(), to implement filters
and freqz() to obtain the filter’s frequency response. As a result, you should learn
how to characterize a filter by knowing how it reacts to different frequence
components in the input.
In the last lab, you learned about both L-point average and nulling filters. Another
very important FIR filter is known as the bandpss filter (BPF). For the res of the lab,
you will learn how to design these filters and how to use them to do certain tasks for
you. One practical example is the dual tone multi-frequency (DTMF) signals to dial a
telephone. Read the following Background section before coming to the lab to
spped up the sign-off process in the lab.
1.2 Background: Telephone Touch Tone Dialing
Telephone touch-tone2 pads generate dual tone multiple frequency (DTMF) signals to dial a telephone.
When any key is pressed, the sinusoids of the corresponding row and column frequencies (in Fig. 1) are
generated and summed, hence dual tone. As an example, pressing the 5 key generates a signal containing
the sum of the two tones at 770 Hz and 1336 Hz together.
FREQS 1209 Hz 1336 Hz 1477 Hz 1633 Hz
697 Hz 1 2 3 A
770 Hz 4 5 6 B
852 Hz 7 8 9 C
941 Hz * 0 # D
Figure 1: Extended DTMF encoding table for Touch Tone dialing. When any key is pressed the tones of
the corresponding column and row are generated and summed. Keys A-D (in the fourth column) are not
implemented on commercial and household telephone sets, but are used in some military and other signaling
The frequencies in Fig. 1 were chosen (by the design engineers) to avoid harmonics. No frequency
is an integer multiple of another, the difference between any two frequencies does not equal any of the
frequencies, and the sum of any two frequencies does not equal any of the frequencies.3 This makes it easier
to detect exactly which tones are present in the dialed signal in the presence of non-linear line distortions. 4
1.3 DTMF Decoding
There are several steps to decoding a DTMF signal:
1. Divide the time signal into short time segments representing individual key presses.
2. Filter the individual segments to extract the possible frequency components. Bandpass filters can be
used to isolate the sinusoidal components.
3. Determine which two frequency components are present in each time segment by measuring the size
of the output signal from all of the bandpass filters.
4. Determine which key was pressed, 0–9, A–D, *, or # by converting frequency pairs back into key
names according to Fig. 1.
It is possible to decode DTMF signals using a simple FIR filter bank. The filter bank in Fig. 2 consists
of eight bandpass filters which each pass only one of the eight possible DTMF frequencies. The input signal
for all the filters is the same DTMF signal.
Here is how the system should work: When the input to the filter bank is a DTMF signal, the outputs
from two of the bandpass filters (BPFs) should be larger than the rest. If we detect (or measure) which two
outputs are the large ones, then we know the two corresponding frequencies. These frequencies are then
Touch Tone is a registered trademark 3
More information can be found at: http://www.genave.com/dtmf.htm, or search for “DTMF” on the internet. 4
A recent paper on a DSP implementation of the DTMF decoder, “A low complexity ITU-compliant dual tone multiple
frequency detector”, by Dosthali, McCaslin and Evans, in IEEE Trans. Signal Processing, March, 2000, contains a short discussion of the DTMF signaling system. You can get this paper on-line from the GT library, and you can also get it at
Figure 2: Filter bank consisting of bandpass filters (BPFs) which pass frequencies corresponding to the eight
DTMF component frequencies listed in Fig. 1. The number in each box is the center frequency of the BPF.
used as row and column pointers to determine the key from the DTMF code. A good measure of the output
levels is the peak value at the filter outputs, because when the BPF is working properly it should pass only
one sinusoidal signal and the peak value would be the amplitude of the sinusoid passed by the filter. More
discussion of the detection problem can be found in Section 4.
2.1 Signal Concatenation
In a Lab #4, you created a very long music signal by joining together many sinusoids. When two signals
were played one after the other, the composite signal was created by the operation of concatenation. In
MATLAB, this can be done by making each signal a row vector, and then using the matrix building notation
xx = [ xx, xxnew ];
where xxnew is the sub-signal being appended. The length of the new signal is equal to the sum of the
lengths of the two signals xx and xxnew. A third signal could be added later on by concatenating it to xx.
2.1.1 Comment on Efficiency
In MATLAB the concatenation method, xx = [ xx, xxnew ], would append the signal vector xxnew
to the existing signal xx. However, this becomes an inefficient procedure if the signal length gets to be very
large. The reason is that MATLAB must re-allocate the memory space for xx every time a new sub-signal
is appended via concatenation. If the length xx were being extended from 400,000 to 401,000, then a clean
section of memory consisting of 401,000 elements would have to be allocated followed by a copy of the
existing 400,000 signal elements and finally the append would be done. This is clearly inefficient, but would
not be noticed for short signals.
An alternative is to pre-allocate storage for the complete signal vector, but this can only be done if the
final signal length is known ahead of time.
2.1.2 Encoding from a Table
Explain how the following program uses frequency information stored in a table to generate a long signal
via concatenation. Determine the size of the table and all of its entries, and then state the playing order of the
frequencies. Determine the total length of the signal played by the soundsc function. How many samples
and how many seconds?
ftable = [1;2;3;4;5]*[80,110]
fs = 10000;
xx = [ ];
disp(’— Here we go through the Loop —’)
keys = rem(3:12,10) + 1;
for ii = 1:length(keys)
kk = keys(ii);
xx = [xx,zeros(1,400)];
krow = ceil(kk/2);
kcol = rem(kk-1,2) + 1;
xx = [xx, cos(2*pi*ftable(krow,kcol)*(0:1199)/fs) ];
2.2 Overlay Plotting
Sometimes it is convenient to overlay information onto an existing MATLAB plot. The MATLAB command
hold on will inhibit the figure erase that is usually done just before a new plot. Demonstrate that you
can do an overlay by following these instructions:
(a) Plot the magnitude response of the 5-point averager, created from
Make sure that the horizontal frequency axis extends from ⇥ to +⇥.
(b) Use the stem function to place vertical markers at the zeros of the frequency response.
hold on, stem(2*pi/5*[-2,-1,1,2],0.3*ones(1,4),’r.’), hold off
3 Warm-up: DTMF Synthesis
3.1 DTMF Dial Function
Write a function, dtmfdial.m, to implement a DTMF dialer based on the frequency table defined in
Fig. 1. A skeleton of dtmfdial.m is given in Fig. 3. In this warm-up, you must complete the dialing code
so that it implements the following:
1. The input to the function is a vector of characters, each one being equal to one of the key names on
the telephone. The MATLAB structure called dtmf contains the key names in the field dtmf.keys
which is a 4 ⇥ 4 that corresponds exactly to the keyboard layout in Fig. 1.
2. The output should be a vector of samples fs = 10000 Hz containing the DTMF tones, one tone pair
per key. Remember that each DTMF signal is the sum of a pair of (equal amplitude) sinusoidal signals.
The duration of each tone pair should be exactly 0.20 sec., and a silence, about 0.05 sec. long, should
separate the DTMF tone pairs. These times can be declared as fixed code in dtmfdial. (You do
not need to make them variable in your function.)
function xx = dtmfdial(keyNames,fs)
%DTMFDIAL Create a signal vector of tones which will dial
% a DTMF (Touch Tone) telephone system.
% usage: xx = dtmfdial(keyNames,fs)
% keyNames = vector of characters containing valid key names
% fs = sampling frequency
% xx = signal vector that is the concatenation of DTMF tones.
dtmf.keys = …
dtmf.colTones = ones(4,1)*[1209,1336,1477,1633];
dtmf.rowTones = [697;770;852;941]*ones(1,4);
Figure 3: Skeleton of dtmfdial.m, a DTMF phone dialer. Complete this function with additional lines of
3. The frequency information is given astwo 4⇥4 matrices(dtmf.colTones and dtmf.rowTones):
one contains the column frequencies, the other has the row frequencies. You can translate a key such
as the 6 key into the correct location in these 4 ⇥ 4 matrices by using MATLAB’s find function. For
example, the key 6 is in row 2 and column 3, so we would generate sinusoids with frequencies equal
to dtmf.colTones(2,3) and dtmf.rowTones(2,3).
To convert an key name to its corresponding row-column indices, consider the following example:
[ii,jj] = find(’3’==dtmf.keys)
Also, consult the MATLAB code in Section 2.1 above and modify it for the 4⇥4 tablesin dtmfdial.m.
4. You should implement error checking so that an illegitimate key name is rejected.
Your function should create the appropriate tone sequence to dial an arbitrary phone number. When played
through a telephone handset, the output of your function will be able to dial the phone. You could use
specgram to check your work.5
Instructor Verification (separate page)
3.2 Simple Bandpass Filter Design
The L-point averaging filter is a lowpass filter. Its passband width is controlled by L, being inversely
proportional to L. It is also possible to create a filter whose passband is centered around some frequency
other than zero. One simple way to do this is to define the impulse response of an L-point FIR as:
h[n] = cos(⇤ˆcn), 0 ⇤ n < L
where L is the filter length, and ⇤ˆc is the center frequency that defines the frequency location of the passband.
For example, we pick ⇤ˆc = 0.2⇥ if we want the peak of the filter’s passband to be centered at 0.2⇥. Also,
In MATLAB the demo called phone also shows the waveforms and spectra generated in a DTMF system.
it is possible to choose so that the maximum value of the frequency response magnitude will be one. The
bandwidth of the bandpass filter is controlled by L; the larger the value of L, the narrower the bandwidth.
This particular filter is also discussed in the section on useful filters in Chapter 7 of DSP First. Also, you
designed some BPFs based on this idea in the previous lab.
(a) Generate a bandpass filter that will pass a frequency component at ⇤ˆ = 0.2⇥. Make the filter length
(L) equal to 50. Figure out the value of so that the maximum value of the frequency response
magnitude will be one. Make a plot of the frequency response magnitude and phase.
Hint: use MATLAB’s freqz() function to calculate these values.
(b) The passband of the BPF filter is defined by the region of the frequency response where |H(ejˆ )|
is close to its maximum value of one. Typically, the passband width is defined as the length of the
frequency region where |H(ejˆ )| is greater than 1/
⌃2 = 0.707. Note: you can use MATLAB’s
find function to locate those frequencies where the magnitude satisfies |H(ejˆ )| ⌅ 0.707 (similar
to Fig. 4).
0 0.5 1 1.5 2 2.5 3
BANDPASS FILTER (centered at 0.4π)
Figure 4: The frequency response of an FIR bandpass is shown with its passband and stopband regions.
Use the plot of the frequency response for the length-51 bandpass filter from part (a) to determine the
Instructor Verification (separate page)
(c) If the sampling rate is fs = 10000 Hz, determine the analog frequency components that will be passed
by this bandpass filter. Use the passband width and also the center frequency of the BPF to make this
Instructor Verification (separate page)
4 Lab: DTMF Decoding
A DTMF decoding system needs two pieces: a set of bandpass filters (BPF) to isolate individual frequency
components, and a detector to determine whether or not a given component is present. The detector must
“score” each BPF output and determine which two frequencies are most likely to be contained in the DTMF
tone. In a practical system where noise and interference are also present, this scoring process is a crucial
part of the system design, but we will only work with noise-free signals to understand the basic functionality
in the decoding system.
To make the whole system work, you will have to write three M-files: dtmfrun, dtmfscore and
dtmfdesign. An additional M-file called dtmfcut can be downloaded from Web-CT. The main M-file
should be named dtmfrun.m. It will call dtmfdesign.m, dtmfcut.m, and dtmfscore.m. The
following sections discuss how to create or complete these functions.
4.1 Simple Bandpass Filter Design: dtmfdesign.m
The FIR filters that will be used in the filter bank (Fig. 2) are a simple type constructed with sinusoidal
impulse responses, as already shown in the Warm-up. In the section on useful filters in Chapter 7, a simple
bandpass filter design method is presented in which the impulse response of the FIR filter is simply a finitelength cosine of the form:
h[n] = cos 2⇥fbn
, 0 ⇤ n ⇤ L1
where L is the filter length, and fs is the sampling frequency. The constant gives flexibility for scaling
the filter’s gain to meet a constraint such as making the maximum value of the frequency response equal to
one. The parameter fb defines the frequency location of the passband, e.g., we pick fb = 852 if we want to
isolate the 852 Hz component. The bandwidth of the bandpass filter is controlled by L; the larger the value
of L, the narrower the bandwidth.
function hh = dtmfdesign(fcent, L, fs)
% hh = dtmfdesign(fcent, L, fs)
% returns a matrix (L by length(fcent)) where each
% column contains the impulse response of a BPF, one
% for each frequency in fcent
% fcent = vector of center frequencies
% L = length of FIR bandpass filters
% fs = sampling freq
% Each BPF must be scaled so that its frequency response has a
% maximum magnitude equal to one.
Figure 5: Skeleton of the dtmfdesign.m function. Complete this function with additional lines of code.
(a) Devise a MATLAB strategy for picking the constant so that the maximum value of the frequency
response will be equal to one. Write the one or two lines of MATLAB code that will do this scaling
operation in general. There are two approaches here:
(a) Mathematical: derive a formula for from the formula for the frequency response of the BPF.
Then use MATLAB to evaluate this closed-form expression for .
(b) Numerical: let MATLAB measure the peak value of the unscaled frequency response, and then
have MATLAB compute to scale the peak to be one.
(b) Complete the M-file dtmfdesign.m which is described in Fig. 5. This function should produce all
eight bandpass filters needed for the DTMF filter bank system. Store the filters in the columns of the
matrix hh whose size is L ⇥ 8.
(c) The rest of this section describes how you can exhibit that you have designed a correct set of BPFs. In
particular, you should justify how to choose L, the length of the filters. When you have completed
your filter design function, you should run the L = 50 and L = 100 cases, and then you should
determine empirically the minimum length L so that the frequency response will satisfy the
specifications on passband width and stopband rejection given in part (f) below.
(d) Generate the eight (scaled) bandpass filters with L = 50 and fs = 10000. Plot the magnitude of the
frequency responses all together on one plot (the range 0 ⇤ ⇤ˆ ⇤ ⇥ is sufficient because |H(ejˆ )| is
symmetric). Indicate the locations of each of the eight DTMF frequencies (697, 770, 852, 941, 1209,
1336, 1477, and 1633 Hz) on this plot to illustrate whether or not the passbands are narrow enough to
separate the DTMF frequency components. Hint: use the hold command and markers as you did in
(e) Repeat the previous part with L = 100 and fs = 10000. The width of the passband is supposed to
vary inversely with the filter length L. Explain whether or not that is true by comparing the length
100 and length 50 cases.
(f) As help for the previous parts, recall the following definitions: The passband of the BPF filter is
defined by the region of ⇤ˆ where |H(ejˆ )| is close to one. Typically, the passband width is defined as
the length of the frequency region where |H(ejˆ )| is greater than 1/
⌃2 = 0.707.
The stopband of the BPF filter is defined by the region of ⇤ˆ where |H(ejˆ )| is close to zero. In this
case, it is reasonable to define the stopband as the region where |H(ejˆ )| is less than 0.25.
Filter Design Specifications: For each of the eight BPFs, choose L so that only one frequency lies
within the passband of the BPF and all other DTMF frequencies lie in the stopband.
Use the zoom on command to show the frequency response over the frequency domain where the
DTMF frequencies lie. Comment on the selectivity of the bandpass filters, i.e., use the frequency
response to explain how the filter passes one component while rejecting the others. Is each filter’s
passband narrow enough so that only one frequency component lies in the passband and the others are
in the stopband? Since the same value of L is used for all the filters, which filter drives the problem?
In other words, for which center frequency is it hardest to meet the specifications for the chosen value
4.2 A Scoring Function: dtmfscore.m
The final objective is decoding—a process that requires a binary decision on the presence or absence of the
individual tones. In order to make the signal detection an automated process, we need a score function that
rates the different possibilities.
(a) Complete the dtmfscore function based on the skeleton given in Fig. 6. The input signal xx to the
dtmfscore function must be a short segment from the DTMF signal. The task of breaking up the
signal so that each short segment corresponds to one key is done by the function dtmfcut prior to
(b) Use the following rule for scoring: the score equals one when maxn |yi[n]| ⌅ 0.71; otherwise, it is
zero. The signal yi[n] is the output of the i-th BPF.
(c) Prior to filtering and scoring, make sure that the input signal x[n] is normalized to the range [2, +2].
With this scaling the two sinusoids that make up x[n] should each have amplitudes of approximately
function sc = dtmfscore(xx, hh)
% usage: sc = dtmfscore(xx, hh)
% returns a score based on the max amplitude of the filtered output
% xx = input DTMF tone
% hh = impulse response of ONE bandpass filter
% The signal detection is done by filtering xx with a length-L
% BPF, hh, and then finding the maximum amplitude of the output.
% The score is either 1 or 0.
% sc = 1 if max(|y[n]|) is greater than, or equal to, 0.71
% sc = 0 if max(|y[n]|) is less than 0.71
xx = xx*(2/max(abs(xx))); %–Scale the input x[n] to the range [-2,+2]
Figure 6: Skeleton of the dtmfscore.m function. Complete this function with additional lines of code.
1.0.6 Therefore the scoring threshold of 0.71 corresponds to a 71% level for detecting the presence of
(d) The scoring rule above depends on proper scaling of the frequency response of the bandpass filters.
Explain why the maximum value of the magnitude for H(ejˆ ) must be equal to one. Consider the
fact that both sinusoids in the DTMF tone will experience a known gain (or attenuation) through
the bandpass filter, so the amplitude of the output can be predicted if we control both the frequency
response and the amplitude of the input.
(e) When debugging your program it might be useful to have a plot command inside the dtmfscore.m
function. If you plot the first 200–500 points of the filtered output, you should be able to see two
cases: either y[n] is a strong sinusoid with an amplitude close to one (when the filter is matched to
one of the component frequencies), or y[n] is relatively small when the filter passband and input signal
frequency are mismatched.
4.3 DTMF Decode Function: dtmfrun.m
The DTMF decoding function, dtmfrun must use information from dtmfscore to determine which
key was pressed based on an input DTMF tone. The skeleton of this function in Fig. 7 includes the help
The function dtmfrun works as follows: first, it designs the eight bandpass filters that are needed,
then it breaks the input signal down into individual segments. For each segment, it will have to call the
user-written dtmfscore function to score the different BPF outputs and then determine the key for that
segment. The final output is the list of decoded keys. You must add the logic to decide which key is present.
The input signal to the dtmfscore function must be a short segment from the DTMF signal. The task
of breaking up the signal so that each segment corresponds to one key is done with the dtmfcut function
which is called from dtmfrun. The score returned from dtmfscore must be either a 1 or a 0 for each
frequency. Then the decoding works as follows: If exactly one row frequency and one column frequency
are scored as 1’s, then an unique key is identified and the decoding is probably successful. In this case,
you can determine the key by using the row and column index. It is possible that there might be an error
The two sinusoids in a DTMF tone have frequencies that are not harmonics. When plotted versus time, the peaks of the two
sinusoids will eventually line up.
function keys = dtmfrun(xx,L,fs)
%DTMFRUN keys = dtmfrun(xx,L,fs)
% returns the list of key names found in xx.
% keys = array of characters, i.e., the decoded key names
% xx = DTMF waveform
% L = filter length
% fs = sampling freq
dtmf.keys = …
dtmf.colTones = ones(4,1)*[1209,1336,1477,1633];
dtmf.rowTones = [697;770;852;941]*ones(1,4);
center_freqs = …. %<============================FILL IN THE CODE HERE
hh = dtmfdesign( center_freqs,L,fs );
% hh = L by 8 MATRIX of all the filters. Each column contains the
% impulse response of one BPF (bandpass filter)
[nstart,nstop] = dtmfcut(xx,fs); %<–Find the tone bursts
keys = ;
x_seg = xx(nstart(kk):nstop(kk)); %<–Extract one DTMF tone
…. %<=========================================FILL IN THE CODE HERE
Figure 7: Skeleton of dtmfrun.m. Complete the for loop in this function with additional lines of code.
in scoring if too many or too few frequencies are scored as 1’s. In this case, you should return an error
indicator (perhaps by setting the key equal to 1). There are several ways to write the dtmfrun function,
but you should avoid excessive use of “if” statements to test all 16 cases. Hint: use MATLAB’s logicals
(e.g., help find) to implement the tests in a few statements.
4.4 Telephone Numbers
The functions dtmfdial.m and dtmfrun.m can be used to test the entire DTMF system as shown in
Fig. 8. You could also use random digits (e.g., ceil(15.9*rand(1,22)+0.09)) in place of 1:16 in
dtmfdial. For the dtmfrun function to work correctly, all the M-files must be on the MATLAB path. It
fs = 10000; %<–use this sampling rate in all functions
tk = [’A’,’B’,’C’,’D’,’*’,’#’,’0’,’1’,’2’,’3’,’4’,’5’,’6’,’7’,’8’,’9’];
xx = dtmfdial( tk, fs );
L = … %<–use your value of L
dtmfrun(xx, L, fs)
Figure 8: Testing the DTMF system.
is also essential to have short pauses in between the tone pairs so that dtmfcut can parse out the individual
In your lab report, demonstrate a working version of your programs by running it on the following phone
In addition, make a spectrogram of the signal from dtmfdial to illustrate the presence of the dual tones.
When you submit your lab report, you must demonstrate your work to your TA. Have your code and files
ready for the demo. You should call dtmfrun for a signal xx provided by your TA. The output should be
the decoded telephone number. The evaluation criteria is shown at the end of the verification sheet.