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( Figure 4 shows the values read from the file named Dsp029.txt and displayed in an improved format.)

In this case, the third frequency in the list is one-half the sampling frequency, which is the folding frequency. The two frequencies on either side ofthat one have values that are symmetrical about the folding frequency.

Figure 4. A new set of frequency values.
Data length: 50 Number sinusoids: 5Frequencies 0.1250.25 0.50.75 0.875Amplitudes 90.090.0 90.090.0 90.0

The program output

The five horizontal plots in Figure 5 show the result of running the program named Dsp029 with the frequencies shown in Figure 4 .

Figure 5. Program output for five sinusoids.
missing image

The center plot in Figure 5 is the sampled representation of the folding frequency.

The top two plots in Figure 5 are obviously the sampled representations of the two lower frequencies specified by the first two frequencies in Figure 4 .

Not so obvious ...

However, it is not so obvious that the bottom two plots in Figure 5 are the sampled representations of the two higher frequencies specified by the last twofrequencies in Figure 4 . They look exactly like the top two plots but in reverse order.

Unable to distinguish ...

Frequencies above one-half the sampling frequency are not distinguishable by viewing the sampled data. In fact, they are converted to lower frequencies bysampling process. The new lower frequencies fold around a point in the frequency spectrum given by one-half the sampling frequency. That is why it is called thefolding frequency. (In 1933, this frequency was named after scientist Harry Nyquist .)

One more example

Let's look at one more example of plotted sinusoids. Consider the frequency values shown in Figure 6 . The second and third frequencies are symmetrical about the sampling frequency. The fourth and fifth frequencies are symmetrical abouttwice the sampling frequency. The first frequency value is the same distance from zero as the other four frequencies are from the sampling frequency andtwice the sampling frequency.

Figure 6. One more example.
Data length: 50 Number sinusoids: 5Frequencies 0.10.9 1.11.9 2.1Amplitudes 90.090.0 90.090.0 90.0

The program output

Figure 7 shows the output produced by running the program named Dsp029 with the frequency parameters specified by Figure 6 .

Figure 7. Program output for five sinusoids.
missing image

The sinusoids are indistinguishable

Although the actual frequencies of the five cosine functions are significantly different, once they are sampled, they are indistinguishable. Thesampling process converts the actual frequencies to new frequencies that not only fold around one-half the sampling frequency, they also fold around allmultiples of one-half the sampling frequency.

Beginning of the class named Dsp029

The program named Dsp029 is provided in Listing 16 near the end of the module. I will break the program down and explain it in fragments,beginning with the fragment shown in Listing 1 .

Listing 1. Beginning of the class named Dsp029.
class Dsp029 implements GraphIntfc01{ final double pi = Math.PI;//for simplification//Begin default parameters int len = 400;//data lengthint numberSinusoids = 5; //Frequencies of the sinusoidsdouble[] freq = {0.1,0.25,0.5,0.75,0.9};//Amplitudes of the sinusoids double[]amp = {75,75,75,75,75}; //End default parameters

Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
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12, 17, 22.... 25th term
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I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
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Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
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I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
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it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
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Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
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ninjadapaul
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Commplementary angles
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Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
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s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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