CS 591 S1 Midterm Exam

Transcription

1 Name: CS 591 S1 Midterm Exam Spring 2017 You must complete 3 of problems 1 4, and then problem 5 is mandatory. Each problem is worth 25 points. Please leave blank, or draw an X through, or write Do Not Grade, on the problem you are eliminating; I will grade the first 3 I get to if I can not figure out your intention. If answers are on the back of the page please tell me so. Circle final answers and show all work. Problem One. (Basic Manipulation of Sine Waves) Consider the following combination of sine waves that produce the signal X shown on the right (and note that the amplitudes are given in relative terms): = X 0.2 = X X = X 1 + X 2 + X = X (A) Give the spectrum of this signal (as a graph of frequency vs amplitude, not a list of triples).

2 (B) Suppose we add X and a signal Y with spectrum [ (2, 0.1, -3π/2) ]. What would be the spectrum of the resulting signal? Solution: X 2 and Y add together: (C) Suppose we add X and a signal Z with spectrum [ (2, -0.1, π/2) ]. What would be the spectrum of the resulting signal? Solution: Z subtracts from X 2 : (D) Suppose we add X and a signal Q with spectrum [ ( -3, 0.2, π) ]. What would be the spectrum of the resulting signal? [You should express your spectrum with only positive frequencies.] Solution: Q = -(X 3 ) so they cancel: (E) Suppose we add X and a signal R with spectrum [ (1, 0.2, π) ]. What would be the spectrum of the resulting signal? Solution: R = -(X 1 ) so they cancel:

3 Problem Two. (Digital Filters) Consider the following spectrum of a random white noise signal, which contains all possible frequencies with equal probability: (A) Suppose you applied a filter to this signal, consisting of F = [0.5, 0.0, 0.5]. Approximately what would the resulting spectrum look like? Solution: The lowest frequency cancellation occurs when 2 samples = half a wavelength, so the period is 4 samples and the frequency is 44100/4 = 11025: (B) Suppose a filter of the form F = [0.5, 0.0,., 0.0, 0.5] were applied to the random white noise signal, producing the following spectrum. What would be your best guess as to the exact form of the filter F? Solution: Using the same reasoning as above, the lowest cancelled frequency is 44100/(2w) for a window duration of w samples. For w = 3, we have 7350 (hm. doesn t look quite right), and for w = 4, we have , which looks right; next cancelled frequency is 3* = , which again looks right. So F = [0.5, 0.0, 0.0, 0.0, 0.5]

4 Problem Three. (Python Implementation) For this problem, you must write a Python function def ampmodnote(f1, A1, f2, A2, h, duration):.. your code here. which produces a amplitude modulated signal of length duration seconds, consisting of a carrier signal with spectrum (f1, A1, 0), a modulating signal with spectrum (f2, A2, 0), following an exponential decay envelope of the form: duration

5 Problem Four. This problem concerns Pure Data, which you explored in HW 03. (A) Bob is a little rusty with Pd, so to review the language he creates a simple cosine oscillator, with a toggle to enable or disable output. He has configured his audio setup but hasn t turned on DSP yet, so while he s confirmed that sound is working, he has no idea if this patch will work or not. Will this patch work? Explain. If the patch does not work, how can he fix it? Solution: The patch will not work because the Trigger only sends atomic values 0 or 1. Thus, the signal osc~ creates will be either a cosine of frequency 0 Hz (a flat line) or one of frequency 1 Hz. This is, obviously, too low for him to perceive, and much lower than the 440 Hz he expects to hear. The obvious fix would be to remove the toggle, but if he wants to be able to enable or disable output, he should create a *~ Object and attach both the osc~ and the Toggle to the osc~, as depicted below.

6 (B) Bob now wants to implement a volume control. He places a VSlider onto to the patch along with a few other objects, but can t remember where to attach it. Help him out and draw the connection from the VSlider to the appropriate Object. Solution: The outlet of the VSlider goes to the left inlet of the *~ Object.

7 (C) In the patch above, does Bob need the +~ Object? What will it do to the input given into its inlets? Should he keep it or remove it? If he should remove it, draw what the patch should look like below (with the connection from the VSlider you added in the previous question). Solution: *~ adds the signal to itself, effectively doubling its amplitude. At worst, this can cause clipping of the signal, so Bob should remove it.

8 (D) Now that you ve fixed Bob s patch, he saves it as mycosine.pd. But there are still issues: 1. There s currently no way to change the frequency of the osc~ Object, or the amplitude of its output without opening up this patch and adjusting the slider. 2. Bob can t use mycosine in another patch because it cannot accept input or return output. Bob knows that he can get rid of the dac~, since he doesn t need audio output from the mycosine itself. What does Bob need to add, change, or remove so that mycosine can accept inputs for frequency and amplitude and output a signal? (Hint: recall the Objects in Pd that allow you to create ports for input and output.) Solution: There are a number of approaches to this problem, but so long as they had the following they received full points: two inlets, one for frequency and another for amplitude frequency inlet attached to left inlet of osc~ Object amplitude inlet attached to VSlider inlet, or attached to the right *~ inlet (omitting the VSlider is acceptable) at least one outlet~ (note the tilde!) for the resultant signal; a problem which outputs two channels, one for Left and one for Right, also received credit. An example implementation is below.

9 Problem Five (Mandatory Essay). We have spent the first section of this course on music synthesis, first exploring various ways of making sounds of various timbres, shaping the amplitude envelopes in particular ways, and finally putting them together into pieces of music. I would like you to reflect on this process in a series of short essay questions. (A) First of all we explored spectral analysis synthesis, analyzing the spectrum of a musical signal and using peak picking to create a synthesized version of the instrumental timbre. What was the point here, why did we pick peaks and not just use the entire original spectrum to create the synthesized sound? Solution: If we use the original spectrum, then---because the Fourier Transform is a lossless translation back and forth between the time and frequency domains---we would simply get the same exact sounds back, in every detail. We were trying to understand the timbre of the instrument separate from the amplitude envelope, the duration, and even the pitch. So using the original spectrum was not helpful. Instead, by picking peaks, we hoped to isolate those components of the sound due to the properties of the instrument, and not accidents of the particular note being played. If we do this correctly, we should get a general spectrum, such as were provided for the bell, triangle wave, and steel string in the function makespectrum; we could then use this to create a synthesized note with arbitrary pitch, duration, and amplitude envelope. (B) We next explored various modulation techniques, including amplitude modulation. We found that if we picked random parameters for the frequency and amplitude of the carrier and modulating frequency, we got rather weird and non-musical sounds. But sometimes we got rather pleasant, musical (if somewhat boring) sounds. When did this occur and why? Solution: As explored in Lecture 6, we know that Amplitude Modulation will reduce the amplitude of the carrier signal, and generate sideband frequencies around the frequencies of the carrier (as shown on right). So for a carrier frequency of f 1, and modulating frequency of f 2, you will get frequencies at f 1 -f 2, f 1, and f 1 +f 2. When these form a harmonic series, the natural result of, say, a vibrating string, you will get pleasant sounds; when it is not even close to a harmonic series, you will get less-pleasant sounds. So for example, if f 1 = 500 and f 2 = 250, you will get 250, 500, 750, which are all multiples of 250, and it will sound reasonably pleasant. If you have a carrier f 1 and one modulating signal f 2, then f 2 should be equal to f 1 /2.

10 (C) A huge problem we discovered is that just synthesizing a spectrum did not produce realistic sounds, because real musical sounds move and change over time, for example due to the characteristics of a string after it is initially plucked. Artificial timbres can be boring and mechanical because they do not change in the same way over time. What was the characteristic of a plucked string that we tried to simulate, and how did we simulate it? Solution: The principal way that a plucked string changes is the roll off or attenuation of the higher frequencies. The higher a frequency is, the more energy required to sustain it, so the higher the frequency, the faster it will decay (if no further energy is applied). This was shown many times when we looked at spectrograms. We simulated this in two ways. You were only responsible for knowing the first one. In frequency modulation, we changed the index or the ratio of the amplitude and the frequency of the modulating signal, that is, as the note played, the amplitude of the modulating signal decayed, reducing the amount of frequency modulation. Since FM introduces many other sideband frequencies, as the amount of FM was reduced, the number of frequencies also reduced. The net effect was to roll off the higher frequencies (and the lower as well, but that wasn t so noticeable). A second way we simulated it (not necessary for this answer, but optional) was to use the Karplus-Strong string synthesis algorithm. This applies a low-pass (smoothing) filter to a ring buffer of samples: we start with white noise (all possible frequencies) and gradually smooth it to emphasize the lower frequencies. The result is a very accurate simulation of the roll off exhibited by the plucked strings for example from the Steel String guitar.