The period of mourning after the death of a parent lasts one year. So I didn't observe the New Year's festival and didn't send New Year's cards.
It's a custom for people in mourning period to send a mourning postcard at the end of the year. In general, receivers of the postcard don't send a New Year's card to the sender.
"Since I am in mourning, I am not sending New Year cards this year"
Kanchu-mimai(寒中見舞い, a mid-winter greeting card) is mailed from Shokan(小寒, minor cold or moderate cold), the 23rd in 24 Solar Terms to the day before Risshun(立春, Spring commences), the first in the terms. Shokan is around January 5, and Risshun is around February 4.
Usually we mail New Year's cards until January 7. The greeting card is used as a reply to a New Year's card after January 7.
a reply to a New Year's card
a substitute for New Year's cards in mourning period