Deputy District Attorney Nicole Rooney told a jury in her opening statement of trial that Zewoinesh Badasso strangled her son in two ways -- with her hands and with a ligature -- on Sept. 7, 2012.
Rooney said Badasso grabbed, squeezed and shook the baby before killing him.
The defendant has pleaded not guilty by reason of insanity.
Prior to the killing, Badasso, 35, told a therapist that she had anger issues, a history of violence and had difficulty controlling her impulses, according to Rooney.
The boy -- named David -- was found about 5 p.m by two passersby in an alley below a third-story apartment where Badasso was staying.
"Please let that be a doll and not a baby," one of the passersby said as they came upon the child, Rooney told the jury.
The person who spotted the body said the mother seemed calm and "uncaring" as they called 911 to try and save the child's life.
Badasso told police she accidentally dropped the baby while trying to open a window.
An autopsy revealed the baby had been strangled, "murdered by his own mother," Rooney said.
Defense attorney Amy McDonald told the jury that Badasso was beaten daily by her father in Ethiopia, leaving her blind in one eye.
Badasso was left with memory problems and tended to "disassociate" herself from her problems, her attorney said.
At 12 years old, Badasso was held down and had her female genitalia mutilated, McDonald said.
Badasso escaped from Ethiopia and came to the United States in 2000, but suffered from major depression and tried to kill herself at least twice, McDonald said.
The defendant had two miscarriages, began to hear voices and was raped by a stranger in 2011, her attorney said.
Badasso got pregnant and was a good mother to David, McDonald said.
The defendant's doctors and therapists disagreed on whether Badasso should be on medication for anxiety, depression and lack of sleep, McDonald said.
Badasso finally took some prescribed medication and went into a "disassociate state" the day of the killing, feeling like she was in a dream, her attorney said.
"It's horrifying," McDonald told the jury, saying her client remember few details about the killing.
McDonald said Badasso felt helpless watching herself kill her child and imagined that he fell out of the window.
Jurors will first hear evidence in the guilt phase of trial. If Badasso is convicted, a sanity phase will follow in which the defense will have to show that the defendant was insane at the time the baby was strangled.